You said that the AC in the living room is blowing warm air. A properly running AC unit should have a 20 degrees F differential between the inlet and cold air vent nearest the AC unit. An IR thermometer like this is very useful for AC, oven, tires, and other temperature measurements.
Something like this:
https://www.harborfreight.com/121-infrared-laser-thermometer-63985.html.
Do you hear the compressor running when you try to cool with the living room AC? It should be a low rumble noise. The AC unit should pull 12 to 15 amps if you have an easy way of measuring this. The dirty little secrets of RV AC's in hot temperature areas is that there is a thermal overtemperature cutout switch mounted on the top of the rooftop compressor exposed to the hot ambient heat before the working heat of the cooling system is added, causing the compressor to cut out when the ambient temperature gets really high. Also, RV air conditioners from fulltiming rigs in hot areas only last about 4 years.
A big help with these problems is IR reflective roof coatings. Henry's Tropi-Cool 887 elastometric silicone roof coating is about 94% IR reflective. An external cooling fan bowing in through the rooftop cover right on the compressor can help, too.