Dennis,
You may well be right, I mean that the house battery is toast.
Let's get something else straight here, the generator only charges the house bank by using the converter/charger the same as if you are on shore power. If that is an old or less capable unit, you might consider replacing it with a modern multi-stage unit. Find out what it is and come back here for suggestions.
Also, you kept talking about the battery as small and singular. If it is a single 12V and you expect to do any dry camping at all, then maybe you should try to replace it with a pair of GC2s. Golf cart batteries are designed to take some abuse. I just replaced an 8yo pair.
Is the coach somewhere that you can get to shore power? If not, you might look into a tiny solar panel to change the house bank while the coach is stored.
Matt