from you last remarks I suspect the problem may be indeed be in the area of the thermostat like the A/C mechanic stated. This requires 12VDC to operate. You need to make sure the 12VDC is present at the thermostat. This 12VDC will come from the 12VDC distribution panel which is feed both by the converter unit and/or the connected 12VDC battery.
Be sure all of the ATC automotive type fuses in the 12VDC Distribution panel are good...
This may have been answered before but are all the ceiling lights and other 12VDC items working now when plugged into shore power...
I also didnt realize this is an older post...
Just my thoughts
Roy Ken