Here's my follow-up...
I bought a Honeywell heat only battery operated TS online. Several that posted reviews said that they installed it in an RV, so that was a good sign.
Got it in the mail just before we were leaving to go to camp, so it got installed right away.
It was snowing, so we spent the day in the TT. I set it at various temps and charted the performance. It would start the furnace at 1 degree below the set point, and stop 2 degrees above it. My thermometer on the other end of the trailer read the same as the digital temp on the unit.
Couldn't be more pleased. When we leave to go hunting, or to town to the grocery store, I can set it at 50 and it will maintain perfectly. 70 when we're inside and 60 at night.
Best of all, the on-off switch is a simple click and I don't feel like I'm going to break it.
The old unit is in the new box under the seat, as a backup just in case.