The DM has the final decision.
The PHB (Chap 9, pag 189, section “Your turn”) says:
Various class features, spells, and other abilities let you take an additional action on your turn called a bonus action.
The fact that some actions are bonus actions depends mainly on your class features: some of them requires a physical effort (e.g. an attack, dashing, hiding), some others just a quick thought or minimal body movement (e.g., cast a verbal spell). In the former case, suffering an internal injury (e.g. a broken rib) makes taking the action more difficult than in the latter.
Since the optional rule does not say anything about bonus actions, It is up to the DM. There are 3 options:
- consider all the bonus actions as “normal” actions, as you called it;
- the optional rule does not say anything about bonus actions, hence they do not fall in this ruling;
- make the character take the saving throw depending on the type of the bonus action.
To clarify the above distinction among bonus actions that requires physical effort and others that do not, down below I list a few examples.
A rouge has the Cunning Action feature that says
You can take a bonus action on each of your turns in combat. This action can be used only to take the Dash, Disengage, or Hide action.
A monk can use its Flurry of Blows feature as bonus action (once the Attack action has been taken)
Immediately after you take the Attack action on your turn, you can spend 1 ki point to make two unarmed strikes as a bonus action.
A druid of the Circle of the Moon can wildshape with a bonus action:
When you choose this circle at 2nd level, you gain the ability to use Wild Shape on your turn as a bonus action, rather than as an action.
Moreover, while wildshaped, the druid can use its bonus action to regain HPs spending a spell slot:
Additionally, while you are transformed by Wild Shape, you can use a bonus action to expend one spell slot to regain 1d8 hit points per level of the spell slot expended.
If I were the DM, I would apply the 3rd option: in case the taken bonus action is mainly physical, then the saving throw must be taken, otherwise you do bonus action safely.