Converter is charging batteries and providing 12 volt power as needed to the RV. The fans run any time it needs to be cooled. It's POSSIBLE that the converter is going bad or maybe it's just working hard in hot weather. The only thing that air conditioner would do is use 12 volt power for the thermostat and that would be a miniscule amount of power.
YOu can check your converter by measuring current across the positive and negative leads AT the converter but this should be done while plugged into 120 volt AC power AND with the batteries isolated. If you test with batteries connected you will read the batteries not the converter.
If you suspect a bad battery (doesn't sound like that's the problem, feel for heat on the sides of the cases. Remove hot batteries from the load immediately. They can explode if they have an internal short and over heat.
Good luck / Skip
2011 F-150 HD Ecoboost 3.5 V6. 2550 payload, 17,100 GCVWR -
2004 F-150 HD (Traded after 80,000 towing miles)
2007 Rockwood 8314SS 34' travel trailer
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