On another thread, an ONAN tech used the term "Load Acceptance" and that's what I was thinking here. I had to replace the carb on our little 4.0 KY genset. Before that, I could start the genset, start the A/C, set A/C so it never cycled (max cold) and the genset would carry that load till I shut it off. If I changed that load in any way (add or remove load) the genset would shut down and not start again till it cooled.
So I wonder, Did this Problem simply Develop, or was there Disuse that might have allowed a little Varnishing to occur in the Carburetor? On our KY it was probably the "idle" circuit in trouble with some varnishing. Counterintuitive, but the Idle Circuit plays a part in loaded operation. I don't know that there are carbs any more sensitive and finicky than on small gensets.
PackPack often mentions Carbon. Our old BFA was one of the ONAN engines with the Carbon Issue. I bought it used and not running, no idea of hours on it, and didn't know about the carbon issue. If I had, I probably would have cleaned it and installed new head gaskets before installing in the coach. Once in, no way was I going after heads when it was running OK. Our problem was a varnished carb (which I struggled with before having it professionally cleaned ultrasonically), AND a weakened Governor Spring.
I wish I had the experience to tell you if going after Carbon or Carburetor is your next best step.