To understand the wiring We really do need a diagram to understand how its wired. Many bikes use the battery as a kind of ballast to smooth out the flow of sparky things. a rectifier may also be present so your lights get DC and not AC from the stator.. I would just grab a new battery as you will need it for items like flashers and horn as they run on DC , unlike lights that can run either.
assuming the wiring has not been butchered Id say the battery is the answer to your issues.
Just had a quick look at 2009 (page 61) that should not be that much different.
http://madeinenduro.com/wp-content/uploads/2014/12/OM_KTM_2009_125_200_250_300.pdfNo. battery has SFA to do with the system. You have a genny that feeds a reg/rec and then on to the accessories and another branch to the start relay and battery.
As I read it, Battery is only there to supply juice for the starter and probably static lighting for current registration requirements on moderns.