I think we're all set...
OK, so I've made some good progress... it's pretty much fixed. My only regret is not listening to Biggrigg prior to taking the PUP in for "warranty" work and paying $150 for them to do absolutely nothing. The one thing I didn't do was test the ambient temperature in the PUP prior to adjusting the screw, but I'd assume it was set to come on at 80, and cut off at around 75 at the coldest... it was very warm in there almost all the time.
So I took Biggrigg's advice and removed the thermostat. I found a Torx T8 star-shaped screw in the end of it. Adjusting this screw clockwise appears to make the ambient temperature threshold go down (get colder in the camper). I kept making random adjustments, but for now I've settled on 1.5 complete clockwise rotations, which moved me from around 75 degrees when the compressor finally kicked off, to around 65 when it kicks off now. One thing I've noticed is that if I remove the cover from the AC, it will get down to around 60 degrees, but when I put it back on it only gets down to around 65 degrees... I guess because of the airflow inside of the box hitting the thermostat. I might crank it down just a little more... I think I'd like to be at around 60 degrees when it cuts off... but maybe I'm getting greedy and just asking for it to freeze up. I'm open to opinions on this, and open to hear about what temperatures others are seeing in their PUPs.
To help others some, I attached some pictures. The wire leading to the round metal piece and the white block is the thermostat. The picture where the blue wires are coming in shows the T8 Torx, which is is on the right-hand side of the white thermostat housing.
Thanks again for the advice guys, I could NOT have done it without you!