IMO, In the owners defense, wearing high heels behind a bar isn't the safest thing to do. If you can't / don't want to follow the rules, maybe it is best that you leave. This is something that seems pretty petty, if I'm being honest. The owner is probably worried about a lawsuit. If you fall and hurt yourself, he could be liable.

Can you maybe try to wear a pair of like combat boots with a small heel (like an inch) - they also have rubber bottoms and may be safer? They're all over the place right now in style, as a sort of meet in the middle type of thing?

If you didn't like the job, I can't imagine why you would stay if this other job was so lucrative. Maybe you could just do both? At this point, I wouldn't quit anything, it's not the easiest thing to find a job nowadays, even in hospitality where there is high turnover.