Each circuit's neutral MUST go to the AFCI breaker
Perhaps you cannot easily see this neutral terminal, but it is on the breaker underneath the hot terminal (not side by side).
You ran those neutrals to the neutral bar the way you do with plain breakers, and that is simply wrong. You seem very fixated on doing it that way, and it is simply wrong. Your AFCIs will never work if you insist on doing it wrong.
Your AFCI pigtails MUST go to the neutral bar
And that is for very good reason. If you have done the above correctly, then the circuit will only work if you connect the pigtails.
You have discovered a glitch in your AFCI where if you disregard both instructions above, the circuit "seems to work". However you have sabotaged the AFCI protection, it does not work at all, and you might as well not have wasted the $40 and gone with a plain breaker. If your solitary objective is to fool the inspector, this might work, but it is asinine.
If you wire the AFCI correctly, and it trips, you have an arc or ground fault. FIX IT.
Guess what. When you apply fault-detection devices to an existing (or new) circuit, it sometimes detects a fault! Goshy wow, that's the whole reason we install them, right?
So, when you wire up a fault detection device correctly and it detects a fault, then most likely, it is exactly what it says on the tin: that type of fault.
So you should start by assuming the fault detector is reporting truthfully, and go hunt down that fault in the normal manner for doing so.
I think when people don't know how to do that, they fall back on blaming the detector. Don't fall into that mental trap. Learn to do the above.
Here are the instructions for your AFCI breaker. I affirm everything ThreePhaseEel discusses. ThreePhaseEel is a master expert at breakers, but you don't need to be one to understand the problem here.
As for the "I don't have any neutral spaces left"....
Consolidate up to 3 grounds per screw, but only 1 neutral.
There are a small number of electricians who group each circuit’s neutral and ground together and stick them under the same screw on the neutral bar. Together. Can’t do that. That’s 2 wires per lug, and they’re only listed for one neutral.
Now if you read the panel labeling carefully, CH panels can have 2 to 3 grounds can go under a single screw, but only one neutral because of NEC rules.
So pull ‘em all out. Clump the grounds 2 or 3 per screw (whatever the panel labeling says is allowed). That will free up a lot of spaces, leaving you enough for 1 neutral per screw, alone.
All the screws need to be torqued to panel spec with a torque screwdriver.
This is a mildly dangerous situation. It happens all the time where a wire on a neutral bar loses contact with the bar, usually because of being under-torqued. If the neutral and ground lose contact together, but still are connected to each other, it will electrify all the grounds on the circuit - touch a light switch screw or an appliance metal case and blammo!
And if you’re thinking “oh no, the neutrals are all too short to reach the breakers, because Mr. Snippy went through my panel ‘to make it neat looking’, and cut all the wires down to barely the length they need to go where they’re going right now”... that happens a lot. You will need to extend the neutrals with wire nuts. This is why Mr. Snippy is flat wrong. Harper’s Law is every hot and neutral must be long enough to reach every space in the panel. You can still have “neat”, but you just have to tuck the extra up somewhere. I like to overshoot the breaker and then come back.