There are some ways of doing it with spells. All of them require the caster to be pretty paranoid, and most of them require the caster to be weirdly paranoid about this one specific crow.
Spells that identify magical effects won’t work
The duration of Awaken is instantaneous and produces a permanent effect, so the awakened creature isn’t magical or under the effect of a spell in mechanical terms. You’ve correctly identified that this extends to the charmed condition that the creature is in: it’s a special case of being charmed which isn’t the result of an ongoing spell effect. Dispel Magic and similar spells would have no effect on the creature’s awakened state, nor its being charmed by you, and spells that identify spell effects (such as Identify) will draw a blank.
Spells can reveal the crow’s heightened intelligence
If the crow is charmed, it could be made the target of Beast Bond. The spell will have no effect, because the crow’s intelligence is greater than 4. Whilst there might be other explanations for the spell’s failure, the caster will likely know that something is up.
Spells can reveal the crow’s ability to speak
Command is the most obvious. Simply compel the crow to ‘recite’ and it will be forced to give away its ability to speak
Spells can directly reveal information about the game world
This depends a little on DM adjudication, but a cleric or similar could just cast Divination and ask ‘Is this crow awakened’? Probably hard to riddle your way out of that one, so likely to get a clear enough answer. This is one of the most feasible strategies, because it would be reasonable for a paranoid caster to ask something like ‘am I behind watched?’ and zoom in from there with subsequent castings. (Commune) is more likely to get you a straight answer (yes/no) to a specific question. Contact other plane would also do it for less pious casters.
Battlemaster Fighter and Mastermind Rogue have abilities which can discern enemy abilities relative to their own. Included here for completeness but they’re not normally available to NPCs and Zizak would have to have an intelligence of less than 10 for this to work.
Arbitrary ‘The 9th level spell Wish could obviously do this’