Huh, the fact that deleting the extra bikes from the menu doesn't fix it is pretty surprising. It's possible that there is some other variables with the bikes that also didn't get reset properly. In general, the way this is implemented seems dumb. I don't know why it isn't just the one item, and then a completely unrelated variable for the bike style.
Deleting all 3 bikes did the job, I just spoke to the bike lady to get the bike back and now she'll swap the skin for me whenever.