I have had the same thing happen to me while hooked up to shore power for an extended period. I have just assumed that the battery was charged and the charger had shut down. after a while, the light was back on.
I would try the voltmeter approach to see if the battery is charged.
I think the generator simply supplies 110V power to the system and transfer switch and the convertor does its' thing regardless of the source of power. I think...
Life is a journey, not a destination.
2005 Georgetown 359ts
Bill & Virginia, Kentucky
Nights camped in 2011...78 , 2012...73,
Nights camped in 2013...123, 2014 ...101