I might be getting a bit ahead of myself, but I seem to have solved my problem. I think I was right that it does somehow get baked into the hardware - powering down my VAIO, removing the battery and putting it back in seems to have made the computer forget its setting, and now when I check the battery indicator, instead of reading 80% (not charging), it says "Charging."
Since this was my primary problem, I don't really need the solution to this anymore, but I thought I would share my solution for anyone else struggling with this problem.
I still would like to know how to get the rest of those options back though!