As even gnu suggested, the best approach would be to have a mains powered trickle charger "permanently" (or semi-permanently) connected to the generator battery so that the battery is always at full charge when the line power goes off. This shouldn't require much more than running a fairly light gage wire pair from the charger to the generator battery (or to the generator "12V charging post?). Most "maintenance trickle chargers" only deliver a couple of amps, so AWG14 or even 16 GA wires should be adequate although I'd probably stuff in 10 GA, or perhaps 12, since derating some for DC current is a good idea but mainly just 'cause I tend to be a little conservative in such things.
As the generator appears not to be intended to be portable, my personal preference would be a plain ol' auto (or deep discharge boat?) battery outside the generator but nearby where it's easier to maintain, on a trickle charger and clipped in parallel with the generator's "built-in" battery. The builtins are usually wrapped up in fancy shrouds and stuff which is one reason they tend to use sealed "no-maintenance" ( = expensive) batteries. The much cheaper "boat/automotive" kind generally require a check on water levels & such at least occasonally, although only a little such attention should really be necessary.)
An external battery could also be used for an "emergency light" that turns on automatically when the mains go down (but should have a separate switch to turn it off if the generator starts without giving you an excuse to go futz with it). Based on some past experience - not necessarily very applicable to the current real world - I don't much care for "automatic stuff" that's supposed to do things in the dark where it's hard to see whether it's actually behaving itself. (Or maybe I just like to make it easy to fiddle with my stuff since I've had so many fun stuff bits to fiddle with over the years.)
If the existing setup doesn't maintain stable charge on the battery, the 2+ years is probably pretty close to expected life for most battery types I'd think would be likely in something like this. The replacement should last somewhat longer if consistent charging can be achieved, although assurances about how long anything will last are cheap and "disposable."