Not a full-timer. The 347THT is one of the lightest toy haulers out there. Just about any diesel truck would pull it. At $20K you are probably looking at 10 to 15 year old trucks.
Stopping is the biggest problem with a large 5th wheel. I don't believe a 350/3500 would do that any better than a 250/2500, especially in older trucks. They both usually have the same brakes. Only the springs and maybe rear axles are beefier.
Also, if you are relying on the truck to stop the 5th wheel you are in trouble with anything short of a very large truck. The trailer brakes need to stop the trailer. Tires, axles, and springs need to be rated to support the weight. The engine and tranny need to be heavy duty enough to pull it up the mountain.
Good luck with your search.