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This may be a battery issue. That would be the first place to check and the battery cable connections. Be sure they are clean and free of corrosion.
If the batteries are over 3 years old, they may be failing. If you check the converter or inverter/charger you can see if that may be the issue. A small digital multimeter is good for this. Be sure you check the ground lead to the batteries. If all that checks out, then maybe the thermostat may be the issue. Let me know the make and model.
Is the light fluorescent style? IF so, it may be giving some feedback. If not, and there is no AC ripple out of the charging system, then the thermostat or the connections are suspect.
Then the thermostat or control module would be suspect.
What is the make and version of the thermostat?
Check the fuse after you remove the cover. See if that is bad or the fuse holder has some corrosion. See what the voltage is on the fuse.
What is the voltage AT the batteries? If good there, follow it to the circuit that has dropped. It maybe a shortstop breaker is failing. Those will open up under heat and then close back up when they cool off. If the light has a short that can do this. I am attaching a photo of a shortstop breaker so yo know what they look like. Usually in the battery control center. https://www.bing.com/images/search?q=shortstop+breaker+photo&id=FF1DB9A9E4D0D1A7540EECAAC90C3BA5FC88C3D3&FORM=IQFRBA
An auto breaker won't have the black button on it.
What is the actual model number of your RV? Let me grab some wiring diagrams.
Is that with the light on? If so, then the problem is on that circuit and it must control the AC thermostat as well. See the attached wiring diagrams. http://www.winnebagoind.com/diagram/1998/126757.pdf
If those share a common connection on the hot lead OR the ground it may be loose or have some corrosion. You can add a "test ground" to that circuit to test that theory. Use a length of wire with an alligator clip on each end.
Sounds like you are getting closer to the source of the problem. So, it appears to be on the positive side and not the ground.
Try to see if you can find the junction of that light circuit and the AC. Look at the diagrams and see if those are sharing a feed. Sometimes you can pull out the light and see that. And yes, that is possible. Bypass it and see if everything works.