I do not see it that way.
The appliance manufacturer has no warranty obligation to make their equipment work with a defective AFCI.
That the unit works with a Square D AFCI is evidence that the other AFCI brands are at fault.
If the unit had excess leakage current to ground, OTOH, and could only be used with a GE AFCI, which does not have a GF trip, that would be a product defect. And the cited warranty policy would apply.
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and if the microwave mfg happened to own SqD, then what?
Actually, since the AFCI is not part of the microwave, it may not violate MM. That said, I'm sure there are other rules that forbid speccing one brand or mfg of parts to allow the microwave to work. As long as the electrical installation is up to code, I do not believe a mfg could put in their instructions "only use SqD AFCI breakers" or "warranty only valid with Romex brand branch circuit wiring", and that there have been no threads here of that type that I can remember tells me the mfg rep who spoke to the OP is dead wrong. They are copping out and putting the burden back on him and the customer, which is sad, but typical.
Personally, I would have posted a recording of that convo and the mfg name, and sent them a letter/email. It is not up to the electrician to play 20 breakers with faulty products, whether it's the microwave or AFCI.
the OP doesnt appear to have ruled out some sort of intermittent short that could be legitimately tripping the AFCI, like a loose wire in a jbox, damaged cord, loose receptacle, etc, so before we condemn the breaker or microwave, perhaps more investigation is in order. Sometimes just replacing the breaker works.