1. As you probably realize, the wall tstat has nothing to do with your problem.
2. LOW gas pressure can cause your problem. Do NOT rely on just the stove burners for your test. RUN the Water Heater on LP and then light the stove burners and then start the furnace. WATCH the stove burners as the furnace ignites and runs and see if the flame diminishes at all. IF it does, replace the LP regulator.
3. If the blower motor is not to correcxt RPM, the unit will still light but will not push enough air to heat the RV. This is very rare for this to happen. If it does, you usually get black soot out the outside furnace vent.
4. It is also possible the HI limiti switch is defective and shutting the flame on and off to quickly, the unit cannot reach operating temp, so it blows lukewarm air.
5. What is the year/Brand Model number of the furnace? Doug