Vehicle Alternator/Inverter system
Can you borrow a current meter and see what the slushy machine is drawing when the compressor kicks in? Then try the same test, using the inverter to see what's tripping it up. Maybe your specific inverter isn't up to the task, but another one might be. Or there may be an issue with the wiring to the inverter (too much IR drop maybe?)

To your thought, they do make 110V alternators for underhood. Example here: https://www.fabcopower.com/generat/bgen.htm  Major drawback of this approach: you have to run the engine at a specific RPM to get the correct 60Hz, so that means something to set the throttle, too. Your customers won't like breathing the engine fumes or hearing the noise.

A sure-fire solution (and possibly cheaper, considering installation) would be a small gennie to run the machine.

EDIT: check the battery condition. If they’ve been sitting unused they may have sulfated and thus will not be putting out enough peak current for the compressor start-up.