If yours is like mine, only the thermostat is on the wall, while the controller is in the air return duct in the a/c unit. Mine, a Dometic, was popping the circuit breaker when the compressor would turn off and right back on. Switched to a new digital controller and thermostat, and walla, problem fixed. If that's what it is, and the labor for changing the controller out sin't too high, I would let the dealer do it. For mine, the controller did not come with instructions and the instructions on the Internet were not clear. You need to be able to read wiring diagrams. My son-in-law was an HVAC trained electrician, so, I had to call him for help. The units are not exactly plug for plug compatible, and the digital thermostats take less wires to control the whole mess. Be careful, on mine the old analog thermostat ran on 7.5 volts (supplied by the controller), while the new digital thermostat runs on 12 volts. But, with the new digital controller and thermostat, my unit is working better now than it has in years.
Oh, and check prices on the Internet. Fortunately, the local dealer, when we were camping, had prices just a little higher than the Internet, so, we went with them. Besides, it was the middle of August, in Georgia, and we really needed the air.
Just remembered, the controllers generally run on 12 volts, so they can still run the furnace, when not plugged into shore power. That's why you still hear clicking. When I had a similar problem to you with a previous trailer, Mine would click but not run, I had a loose 120v wire connection behind the electric converter.
