but hidden stuff in Louisiana is ultra vires,
This is a good point, though it may require a good local lawyer.
A friend of mine has had a lot of success putting up external antennas in his HOA. He doesn't have a 70 foot tower or anything, but he does have a good lawyer -- and, several antennas.
In Arizona, at least, it appears that any regulation that does not plausibly have to do with protecting property values is "ultra vires" -- beyond their legal power. They can put "no indoor antennas" in their regulations if they like, but legally, (if I go by my friend, IANAL etc) it is just a bluff. They don't like to pay legal fees, either, especially futile ones.