We've been through this recently for Florida. Whether used on the road or not, travel trailers in Florida are supposed to be tagged. If you want to call it a manufactured home, and pay real property tax, some county assessors might cooperate, but they aren't really supposed to, Florida law gets somewhat specific about what is a RV and what is a manufactured home.
To me, it would make sense to tag it in Florida. I don't know the situation for Virginia, maybe it would be cheaper, but even so, Florida also gets antsy about vehicles permanently in the state that are tagged elsewhere. They are tolerant about enforcement for long-term visitors, but there are limits to tolerance.
In our Michigan snowbirds family it has been vehicles going back and forth Michigan to Florida, tagged in Michigan. Vehicles staying in Florida get tagged in Florida.
I could get tricky, and tell them a travel trailer is "tagged in Oklahoma." Then show them the letter from the Oklahoma Tax Commissioner that says Oklahoma does not require personal use trailers to be titled or licensed. But it is safer to buy the optional tag, it doesn't cost all that much.