FW5000 of that vintage, your generator shutting down is a likely different problem from whatever battery problem you might have. The only connections from house battery to generator are a heavy cable to provide starting current and a lighter wiring to the remote switch and hours meter, if you have that. If you can start the generator, those connections would appear to be good.
The generator will shut down for low oil supply, or if it detects a problem with the 120V circuits that could damage the generator (e.g. a short circuit). I don't know whether the generator checks itself for overheating; if it does, that could occur after running for a while.
