I think your drive will be a strain. What is the rated towing cap on your RS? The Trooper II has got to weigh as much as your RS tow car. No big passes to drive between AZ and Sugarland, but Albequerque is about 4000ft IIRC. Assuming your RS is manual, I'd mainly be concerned about braking and trying to execute panic manuvers. I'm not too happy with towbars either; I'd think about maybe renting a dolly type trailer. Uhaul, as sorry as they are, would fit you up with brake lights in that case. I used to haul my '86 Trooper on occasion on a 16" flatbed trailer with my 2001 Trooper. The trailer adds 1500 lbs to the towed package, but it has electric brakes and puts me right at towing cap. Still I wouldn't want to haul it to Moab from here (I'm approx Denver, CO). I MIGHT, but I wouldn't WANT to: there's a four mile downgrade coming into Denver on I-70 from the west. There is a runaway truck lane about 1.5 miles from the bottom. You probably don't want to use it unless you have to. I'd put a tranny cooler on too, before attemting it.
What do you think about renting a trailer? Check round trip rates; It may be as cheap as one-way, and could give you some insight on the challenges involved as you drive over. You want heavy duty loadbinders holding your "new" Troop down in that case.
I bet that F150 has more horses than your RS but you'd still have braking issues. And it's probably lighter than your RS, especially over the rear wheels.