Puchs come 6v not 12v. Treats has 12v coils you can buy to remedy that situation though. I'm working on a similar project and have found that a little 4amp 12v battery will just barely squeeze under your side cover.
Running a battery is somewhat of an obstacle though, because your electrical system is AC and batteries need DC current. To convert the AC to DC, you need to get a diode bridge rectifier.
You plug one of the AC (~) ends into your power coming from the 12v coils, and the other AC end into the frame. Then you should get DC + and - from the DC ends, which you can run through a regulator into your battery.
Once you have 12v power running though, running a 12v motorcycle taillight should be possible.
Good luck dude!