Mine is hooked up the same way, when shore power is connected, the battery gets charged.... when home, it is always plugged in, so I don't take the battery out in the winter, just keep it charged...
I've done this on my scooter as well, and the 2002 oem battery still works. But, that one I'm going to replace as a cell could short out while I'm away from home, and 7 years is good enough.