gnu, check out motorcycle and lawn tractor batteries as they may fit. They won't have much reserve capacity but if maintained at a proper and constant float voltage they will probably supply all the kick that you need. There are also batteries designed for emergency lighting that supply 6 or 12 volts in a smaller package. You should find them at most electrical supply stores. Canadian Tire offer a 15 watt solar charger, often on sale for less than $100, but a 2 amp trickle charger is much less if you have a place to plug it in. I would be tempted to shitcan the Honda battery and extend the wiring to an external one. If you want the generator to be portable you could always use the manual start.
Properly maintained lead-acid batteries should last for much longer than 2 years if they are not deeply discharged and the fluid level is maintained above the plates.