In some set ups a house battery acts like a "filter" for power. When my house batteries not just went bad but were disabled (they cracked open), I had no 12 volt nor 110 either when plugged in or, on generator. With a new battery power was restored.
If your generator starts using the house battery, a bad one may cause the generator to turn over more slowly. (Although with the truck engine running you should have full cranking power). However, I can't think of a reason why a bad battery would cause the generator to eventually stall out unless maybe there was a heavy load e.g. via a short or something like that.
Note that when exercising your generator once a month or so which is important to do, you should do so with some load on it e.g. the air conditioner. I don't know if running the generator with no load would cause it to quit after a while.
It sounds like you need a new battery anyway (be sure to check the battery water first) so get it and the generator should crank more strongly. If the generator still stalls out when running, with some load on it, then you might have a different problem.
