I doubt that its the ATS because it IS effectively switching from generator power to shore power, and your equipment works fine on shore power. It COULD be the relay on the generator side of the ATS, but even then, thats doubtful.
It sounds more to me like an issue with the generator itself. Most likely dirty slip rings are causing the voltage regulator to work too hard, and its shutting down. I had the same issue happen to me 2 weeks ago. I know its not my voltage regulator as I just bought a brand new one from flight systems and installed it.
Went out with my 'Slip stick' also manufactured by flight systems, started the generator, cleaned the rings, restarted the generator to reset the voltage regulator, never had a problem since.
http://www.flightsystems.com/rv-generator-controls-repairs/slick-stick-slip-ring-cleaner.html
Thats where I bought it from. Reliable seller, well known.
My generator is a 5.5k onan marquis. as well. I have all too much experience with this unit.
This is the quickest and easiest thing to check, and it requires to disassembly of anything. Try it first!