Running the generator should be the same as connecting to shore power... Make sure you do not have any BATTERY DICONNECT SWITCHes engaged. Always handy to have a DC MULTIMETER around the RVs and in this case you should read the DC VOLTAGEs at the battery terminals. A fully charged battery should read 12.6 - 12,7VDC without the generator or shore power connected.
Switching to GENERATOR or SHORE POWER your battery terminals should jump to 13.6VDC or higher depending what charging mode your on-board converter/charger is doing.
We take our batteries very seriously and if they ever display DC VOLTAGEs at 12.0VDC or lower we know we are experiencing battery charging problems. Batteries that get discharged below 12.0VDC (appx 50% charge state) and not get re-charged back up to at least their 90% charge state will not perform well and in most cases will be required to be replaced. We can't afford buying batteries all the time...
IN your case if you are not seeing 13.6VDC or higher at your battery terminals when on Sore Power or Generator then you need to start looking for the Battery Disconnect Switch setting, in-line fuses or 12VDC circuit breaker between the converter/charger and the battery terminal, poor battery connection connection, or in the worse case a bad battery. Check the REVERSE POLARITY fuses in the Power Distribution Panel. Sometimes the batteries will boil out their fluids from being over charged and this also leads to battery failure.
I lost my first new battery in about three months of use before really starting a battery maintenance plan. The remaining three batteries I currently I have on my RV setup have been performing great now since 2009. Lessons learned for me...
Roy Ken