Generally car haulers your looking at come in 7000 gvwr with tandem 3500 lb axles with brakes.
The trailer may weigh 1700-1800 lbs plus the 3800 lb car = 5600 lbs so its well under the 7000 gvwr.
My 10000 gvwr car hauler has tandem 5200 lb axles/brakes but weighs 2520 lbs. Unless your hauling something that weighs in the 7500 lb range you may no need it. However 5200 lb axles times 2 = 10400 lbs of braking is a big deal.
Tires will be your biggest concern.
The 7k trailers usually come with 15" 5 or 6 lug wheel. If you go this way I would have the trailer mfg mount Providers.....Goodyear Endurance.....Carlisle HD ST in that order. They will cost you extra but a much better tire down the road.
The 10k trailer usually come with 16" 6 lug wheels. If you want the best tire for that size trailer pay extra and get the Bridgestone R-238 commercial grade all steel ply carcass tire. We run these 60k plus miles with no issues.
Or if you choose to stick with a ST tire use the same brands I mentioned above.
The trailer dealer will tell you their off brand maypop ST tire is good or better however having placed many trailers is service I and my drivers found out they weren't.
I see your sometimes from Texas. Tx abounds with these type trailer mfg and their dealers.