You dont want to go bigger than the 4x8 U-Haul trailer. If it's just a short haul- like a few hundred miles, I would not worry about a tranny cooler. If it's a longer haul, a tranny cooler is a good idea. U-Haul does install them, as does dealers & tranny shops. I would try a quality independant tranny or repair shop first if you dont want to DIY. Have a good trip!
Joel