"KD4UPL and BFJ13: you recommended a adding a portable charger. Wouldn't I need to be plugged in to shore power to use one of these?"
The charger has battery clamps and a 120v cord to plug it in. With your generator on, you have your own 120v to your receptacles inside and usually one outside. Clamp the charger to the battery posts and plug it in (extension cord if necessary)
Do replace your battery with an AGM like Mex said. It sounds like you have an ordinary starting battery in there, not even "marine/RV" type.
Here is the charger I meant. It would do well with the AGM too. You should be able to keep your gen time to two hours a day with one of these.
http://www.boatandrvaccessories.com/black--decker-vec1093dbd-12-volt-40-amp-battery-charger.html
