Ozeri Brezza II turns on, oscillates but won't spin. No motor sound.

Sorry about the back problem - they're never worth having.

To be clear, so this is normally mains powered? And you have an inverter which plugs into a cigarette lighter socket and has a normal (110V if you're in the US) mains socket on the other end?

According to the Amazon page you helpfully quoted, this fan can take up to 55W, which I understand you've checked is within the capacity of your inverter. But I suspect the problem might rather be that the inverter doesn't produce a pure sine wave output, i.e. instead of alternating smoothly in direction like a pendulum as mains power does (roughly), it alternates in a stepwise fashion, euphemistically called "modified sine wave". The fan motor may not like this. If it still oscillates and responds to the remote then the whole fan isn't toast - perhaps the smaller motor(s) which create the oscillating motion are more forgiving. If you're lucky there could be a thermal fuse that's blown, but if you’re out of luck it could be the speed control circuitry that's died.