A travel trailer works if you pay someone else to haul your horse! But, if you have to haul your horse yourself, it is not really an option (unless you make 2 trips to the show, have 2 trucks and a significant other that will haul one, or something similar.)
There are some REALLY nice LQ’s out there but they will cost 6 figures. I agree that the RV is a great option if you don’t mind paying someone to haul your horse. But, you are going to be spending a lot of money on hauling fees, RV hook ups at shows, etc. If that works for your budget, great!