Certainly in older PUPs, such a rig exists, because we once owned one we towed with a minivan. The only time the van struggled was on the long, fairly steep climb from Rapid City up into the Black Hills - but we still made it.
In order to keep the PUP as lightweight as possible, you'll need to opt for one that's relatively small and has no shower or toilet, no built-in furnace, and few options. (Though for where you're going, an a/c sounds like a necessity.) Travel with the water tank empty as often as possible. (If you can find one with a cabinet that can store a portapotti, you'll at least minimize those hikes to the bathroom in the middle of the night.)
Good luck!
