Hmmm.....
It does sound like you are in a pickle. It is doubtful that the bank/appraiser will reconsider his/her value (regardless if they should or not) and you are left with the choice of going through the permit process retroactively or paying for PMI which might not be necessary if the appraisal considered whatever contributory value the garage conversion has in the market rather than just as a garage.
The "prudent" thing to do is to investigate the jurisdiction's process in obtaining a retroactive permit. That is a local issue so unless an appraiser is familiar with your specific market, there's not much more to add on that point (but I will anyway below!).
My general experience is this: If garage conversions are permissible and one can get a permit, the jurisdiction is usually amenable to retro-permitting the conversion. You can find out what the requirements are without giving away your address; compare what they are vs. what you have and decide where to go from there.
I also work in markets where the requirements are to provide an alternative covered parking structure if the primary garage is converted. In those markets, if one converts without permits and does not provide for the additional covered parking, the city will require the garage to revert back to its original configuration.
You can see how the value of a non-permitted conversion would be different in the two different scenarios:
In the one where it isn't a big deal, the value (in the market) may be similar to GLA.
In the one where it is a big deal, the value could be negative: build the second structure (if there is room on the site) and get the conversion permitted, or lose the living area the conversion offers and pay for converting it back to the original garage.
Hopefully, you'll have an option that allows you to keep the converted area with a minimal fuss and get your home valued to remove the PMI!
Best of luck to you!