I've added insurance for our golf cart since some parks require it, and it's probably a good idea since "we" are a sue happy nation.
You might ask the park if they'd consider letting you use it if you have insurance, and would be willing to sign a liability waiver, releasing them from any damages.
If they still say no, consider going elsewhere. We have PLENTY of great campgrounds in FL. Even state parks now allow them with reasonable exceptions.
A private park can allow or prohibit as they wish. ADA has nothing to do with it. The term "GOLF CART" and "ELECTRIC PERSONAL ASSISTIVE MOBILITY DEVICES" actually have legal descriptions in the FL statutes. They are not legally the same thing, though many might use it as such.
Good luck...