You can do whatever you like. The easiest course of action would be to buy a new AGM battery for $75, install it, slap some plates on the bike and enjoy it for a while. Switching to a solid state regulator requires some wire splicing, not hard at all.
Get the bike tuned up properly and put some miles on it. I registered my bike a few years ago, my insurance has stayed at <$120 for the year and $80 for the registration fee.
Knowing how cheap these bikes are to register and insure, its just easier IMO to get the bike road legal and have all that (paperwork) taken care of. You really don't want to be working on your bike come the next couple months when its really nice out, TRUST ME.
Aside from a tune-up, your bike looks like its fairly complete and only needs a few things to be road worthy. enjoy it and plan as you ride it. Do some reading and get to know the bike.
The AGM battery will be fine with a mechanical regulator, unless the regulator fails. The previous owner could've changed it to a solid state too.
The major thing with our bikes is they need a good battery to run. The bike will not start without a battery and these bikes don't produce a whole lot of charging voltage to keep the battery charged while at a stoplight with your lights on. Like I said, I ride with a 4-cell battery and im risking it, but I installed a volt meter on my bike so I know whats happening.