All the thermostat does is connect RH to white to run the heat or RC to yellow for cool. It also connects RC to green to run the fan. You should be able to short those pairs of terminals to bypass the thermostat. I'm assuming the wiring is fine since it used to work ok. The OP was asking how to bypass to test the heater and A/C.
The only difference between residential and RV use is that residential uses 24V AC and RV uses 12V DC. The thermostat uses relays to connect "incoming" power (RH/RC) to whatever it needs to run (fan, heat, cool).