The condenser fan is most likely of the induction type. I 'think' the compressor will fire up, everything else should be OK. But the stuff that gets hot probably wont be cooled, and the stuff that gets cold probably wont blow cold.
Some of the window AC units use BLDC motors for the circulation fans. I don't know for sure about the compressors on the typical ones, but at one time there was a place advertised on DIY Electric Car forum that sold BLDC type compressors, as part of kits to replace the belt-powered compressors in converted ICE cars.
I've also read a couple years ago (on DIYEC, I think) that a very few cars use BLDC compressors already, powered by the system rather than a belt. I do not remember any details, though, and could be completely misremembering in my flu fever.
As for the circulation fans, even if they *are* induction, it'd be possible to replace them with something else, if you were up to it.
I've read some of them use switched reluctance motors, and the electronics driving that could probably run on DC. But there may be other stuff in it that needs AC. You could maybe ask the manufacturer?