TL; DR: Immersion on 3 levels is probably your best option, as it largely outweighs the abilities you will lose as a paladin given your role in the party. Warlock 3 gives you two additional level 2 spell locations (renewable per break) that can be used to hit or heal yourself, without costing you the ability to cast level 5 spells. The celestial pact more than makes up for your loss of punching power on the hands, and Chain Pact is probably your best option because it allows you, through summoning The Everyday Gift, to maximize every bait you cast to heal yourself. , making you a great tank.
People place a lot of value on Warlock 2 because it gives you access to invocations. If your goal is to become an eldritch blaster, that's all you need, that is, access to eldritch blast and agonizing explosion. Similarly, Warlock 4 means you do not lose ASI, which can be very valuable in cases where multiclassing puts you in a difficult position with respect to ability scores. The two do not apply so much to you because you're probably not going to spam the breath of eldritch considering your role within the party, as well as the fact that paladins usually possess high charisma scores. The warlock 3, on the other hand, is not generally considered very useful, the advantage of the pact being generally wasted during the multiclassing. These are generally useful for lower-level characters, unless you "resize" them with invocations at later levels.
However, in the case of the paladin, especially the one who enters the warlock class at much later levels, I think Warlock 3 is probably optimal for many reasons. You also mention the fact that you are both the tank and the support of your party. So we'll compare what you win / lose by multiclassing by taking a closer look at the following subsets of your abilities: care, casting spells, offensive and defense, and analyzing your different options regarding UPS, Covenant benefits and the invocations.
The healing light function of the celestial protector compensates very well for lost hands. As long as your healing abilities go away, you do not lose anything by multiclassing into a warlock, except that you can not use all your healing in one shot if the need actually feels (put your hands on you and the healing light should be used separately). This feature can also be used up to a distance of 10 meters. This means that, unlike Lay on Hands, you can prevent an ally from dying without having to move, potentially avoiding opportunity attacks.
Since the healing amount of the healing light is 1d6 + 1d6 per level of warlock, it gives you a slight jump of healing at 1st level (7HP on average) and even at 2nd level (10.5HP on average) . A low of 3 levels (average of 14HP against 15) and 4 levels (average of 17.5HP against 20) represents slight nerves to your healing abilities on average, but not significantly.
If healing is your main concern, no matter how big your Warlock dive is, although you get a little more healing potential with a dive of 2 levels or more, because of the second Warlock spell location that could be used to heal more wounds come out. Some invocations (as we will see later) can also turn this into a huge buff for your healing abilities.
A four-level dive costs you the ability to cast 5th level spells, which is a heavy price to pay if you only win one spell (of which you already know 4) and a 2nd level spell known. Since you are the only divine member of the party, the inability to cast 5th level spells means that no member of your party can, for example, lay dead to death.
If you want to maximize casting abilities, I recommend you dive 3 levels into the Warlock. This allows you to keep access to level 5 paladin spells and give you 2 additional Warlock Level 2 Renewable Spell locations. The spells you get from warlock can also be very support oriented. Spells such as Invisibility and Spider Climb can give your thief a big advantage when you slip, for example. Understanding the language can give you an edge in social challenges. Hold Person can be an excellent crowd control. Etc.
Basically, the multiclass in the warlock gives you more known spells, and many of them would not normally be available to paladins. As far as spellcasting is concerned, a 3-level dive is a bonus for your versatility and spellcasting potential, although this somewhat delays your progress.
In addition to delaying the moment when you get an improvement in Divine Strike, which is a very good increase in damage to Paladins, you will not lose anything in the long run. Given your status as a main tank rather than a damage provider, this delay should be acceptable.
In fact, multiclassing in the Warlock can actually increase your damage in situations where you can not immediately enter the melee zone, as the Holy Flame and Eldritch Blast are both very decent. This means that you do not have to waste stock to put away your sword and shield and draw a weapon from a distance: you can simply use these traps while closing the range. In addition, if you take the agonizing breath invocation, Eldritch blast damage may actually compete with your usual melee damage.
The loss of the extended reach of your aura is not as expensive as it looks. In general, remote characters will probably be more than 30 feet away anyway, which means that extended range only makes it easier to position other melee characters in your group. Nevertheless, it is an ability that you acquire very late, and the melee characters in your group should already be used to the required positioning for your auras. So, although it's a very good buff, losing it does not suddenly break the paladin class, because your party can still take full advantage of your auras, as they have done for over 12 levels.
Apart from this, the only defensive disadvantage is, as stated above, the inability to use 100% of your healing powers in one action if you need it. Otherwise, you still care a lot of HP, some of them are just a little more random.
On the other hand, having access to Agathys' armor by multiclassing greatly increases your defensive powers by giving you extra health and dealing damage to enemies who hit you. Since you can cast Agathys Armor using your paladin spell slots, this can mean an additional 20 or 25 health for combat, as well as potentially more than 50 damage to enemies. Basically, this spell alone is a better use of your spell locations than hitting it, as it does more damage in addition to providing an excellent defense. Agathys Armor requires no concentration and lasts an hour, which means that you can even combine it with a mirror image, also available to warlocks, for further defense. Or maybe with a shield of faith.
Anyway, you will lose the class function Emissary of Redemption, but a proper application of Armor of Agathys will provide you with very similar benefits for a while (temporary PV instead of resistance and damage). inflicted on your attackers), all without losing the benefits if you actually attack them during the fight (unlike Emissary of Redemption).
If your campaign actually reaches level 20, the loss of a single UPS should not be as important. At this point, you should already have the feats you want and 20 in your main statistic, and probably 18 or 20 in your secondary statistics. As for the aura buff, that's fine, but it should not break your build if you do not have it.
This one is a little tricky here. Basically, as you do not take more than 4 levels in the Warlock class, none of these abilities expands significantly. You mention the pact of the blade, but that would just give you the ability to store your weapon (probably already magic) in an extra dimensional space. Pact of Blade is generally useful for single-class warriors because it is the only way to get an additional attack (5th level invocation) and additional charisma-based damage (12th level invocations). You have already received features similar to those of a paladin (at the 5th and 11th levels respectively), so the pact of the blade will probably not bring you anything special. Similarly, since your client already gives you 2 extra cantrips, the book pact does not do much for you unless you really want a specific cantrip of a different class.
I would personally suggest going with a pact of chains, which would allow you to get a cool familiar like a pseudo or a sprite (assuming you do not want more perverse pets). In particular, the pseudodragon has a blind sense, which allows it to easily detect invisible creatures within its limited range, as well as having an advantage over virtually all perception controls (sight, hearing and smell). If your DM allows it, you can also potentially harvest his poison. The sprite is also interesting because it can fly and become invisible at will, allowing it to track for you.
More importantly, Pact of the Chain is especially valuable for its access to invocations …
Since you can not hit with your fingers, maximizing your bullet damage is not necessarily what you want to do. You only receive two invocations, so you need to count them as much as possible.
As you have access to Xanathar's Guide to Everything, the gift of living beings is a very interesting option. It requires the Chain Pact feature, but allows you to maximize dice rolled for healing while your pet is within 30 yards of you. Light of healing? Maximized. Heal wounds? Also maximized. Take a short rest? YES! So, yes, this invisible sprite that no enemy really knows, is like a real guardian mini-angel sent by your god to make you invincible.
If you finally take the Blade Pact, the Summoning of the Improved Pact weapon allows you to use your weapon as the target of your Warlock spells. However, you are more likely to use these spell sites to hit or heal yourself by using healing wounds (which you already know as a paladin), which means your sacred shield already covers it. most of it for you.
When it comes to support abilities, some invocations can give you an expansive dark vision that even pierces magical darkness, the ability to detect magic at will, to read any language, to disguise you as you please, to talk to through your pet, etc. What you want to do has many interesting options. However, if you decide to choose chain pact rather than blade pact, Voice of the Chain Master presents some interesting strategic applications because it allows you to send your invisible fairy scout. with the thief and discuss your plans as if they were always with you.
If you dive into a warlock, you must always take at least 2 levels for additional slot housing and access to summons. Losing an ASI is not fun, but you will gain a lot more power than it does not matter. The Heavenly Warlock associated with the Chain Pact makes you a great tank, allowing you to maximize every dice you cast in self-healing, making it a very attractive 3-level dive. Diving 4 levels in the Warlock does not give you much in regards to Warlock abilities, even though your ASI is recovered, but it costs you level 5 paladin spells, which means the absence of similar resuscitation spells.
All things considered, I think that a 3-level immersion in the Warlock is your best option. The only thing you really lose is the extended auras and the delayed access to certain abilities. On the other hand, the appropriate pact invocations / benefits will greatly enhance your healer / tank abilities, which will more than compensate for this loss.