Wow, you've really tried a log of great ideas. I have a couple more ideas for you:
Put one of those non "portable" seats in a backpack and take with you. I am referring to the kind you probably use at home, the hard ring that goes directly onto the toilet and makes the seat smaller. Your portable dora ring might feel floppy and not secure enough? I did that for a while with my daughter who was scared of the size of the toilet opening.
Have your daughter activate the flush (standing, fully clothed, stall open (in the big handicap stall) Maybe this will acclimate her.
Or, just simply go and visit the public restrooms at every possible chance. Especially when she does NOT have to pee. I know its a pain in the butt, but (lol) it might get her used to being around toilets/flushing. You could even make an event out of it. Find a public bathroom that you can stand to be around for some time and simply hang out in it. Don't worry, people will come and go, they won't realize what you are doing. If you're nervous, go stake out your claim on the biggest baddest handicap stall you can find and hang out there. Add in the occasional flush or you sittng on the potty. Be funny when it "accidentally" flushes with you still on it.
If this doesn't work, keep a port a potty in your car (with a roll of leak proof liners. Not ideal, not convenient, kind of gross, but holdig pee for long periods is worse.
Good luck!
Oh, and although I understand why you did it, the one thing I would not do again it timeouts. It's not misbehavior, its fear.