How can I resolve tripping problems with CAFCI?
We can divide this problem into two cases; namely, "instant trip" circuits and "device trip" circuits.  (There are also "any device trips" circuits, but you do not have one of those.)

Circuits that trip a CAFCI instantly when the breaker is installed and switched ON without loads connected/switched on.

This is a clear sign of a wiring flaw in the circuit: either there is a parallel arc present due to faulty wiring (rare), or more likely, a ground fault is present in the circuit, also due to faulty wiring or a mis-wired circuit.  

Examples of the latter include:


landing the circuit neutral on the neutral bar instead of the breaker
cross-connecting the circuit neutral to another circuit's neutral
grounding the neutral at some point downstream of the main panel (this is most commonly the fault of bad subpanel installs, but can happen if someone does something wrong in a box as well)
trying to put a single pole breaker on a multi-wire branch circuit or shared neutral run


Helpfully, you can glean diagnostic information from your breakers when troubleshooting this.  As per this Square-D instruction sheet,


Turn the culprit AFCI OFF
Hold in the TEST button (white button) on the culprit AFCI
While holding in the TEST button, turn the culprit AFCI back on.
If the culprit AFCI trips instantly, it is sensing a ground fault (such as a shared or miswired neutral, although damaged wiring cannot be ruled out).  If it delays a couple of seconds and then trips, it is sensing an arc fault (damaged wiring).


Circuits that trip a CAFCI when a specific device is switched on or plugged in

This is indicative of a faulty device -- in your case, a garbage disposal.  Some older disposals use brushed "universal" motors, and these can produce more severe arcs if the brushes are worn, which could trick the AFCI into tripping.  Having the motor rebuilt with new brushes would fix it in this case.

If your garbage disposal is a newer model using an AC induction motor, though, it is likely simply either a victim of water ingress, or just plain faulty; in either case, it's time to replace the whole disposal.