Hello Mezzo - Welcome to the Forums!
We've had this question come up in the past. There are always professional transporters who will move RVs from one location to the next. I'd probably contact an RV dealership and ask for references.
That's not really your question. U-Haul and other big truck rental companies don't make it easy. Fitting a 10K pound capacity shank and 2 5/16" ball is not in their program. You're asking a tough question. A medium-large SUV or a 1/2-ton truck would fill the bill. Would it have a 2" receiver installed? And then you'd want to install a brake controller for the trailer and that may be more than a rental company would want you to do.
Guess I'd throw the question out to anybody who has tried this. More responses?