Exercising is what generators need - running it once in a while, so that stale and dried fuel wouldn't cr-ap the carburetor and fuel lines.23 extra pounds for a DD motor & long range battery is bad enough. Besides lead acid batteries, in the "maintenance free" variety, sulfate up and become useless if not exercised some large percentage of capacity several times a month. I'm hoping lithium ion batteries don't have the same syndrome.
Led acid batteries, whether it's wet or AGM, don't need this kind of exercising. But they need to be 100% charged most of the days. Failing that, they need what is called "equalizing" - bring it to 100% and then keeping it for an hour at 15.0-15.5V, once every couple of weeks. This roughly can be compared to balancing in Li-ion cells.
AGM batteries need equalizing too, but for this group the procedure is not written, I think only the manufacturer of high-end boat batteries Lifeline have this procedure in the manual.
Li-ion batteries don't need to be charged 100% - they last longer if charged to 90%. Similarly to led acid, they need 100% charge once in a while - I think this is not a result of some inherent chemistry laws, but rather flawed BMS in many batteries that don't do balancing until cells are 90-95% charged.
Yes, car starter battery is a poor choice for battery bank. Better consider RV deep cycle or "marine" battery.