Your SIII already has the wiring in place for a trailer brake controller, which is an important component of the system. To be sure that your SIII is ready to haul trailers with brakes, look for a trailer brake module in the dashboard. If it's missing, you'll need to have it installed.
I had a similar issue with battery drain. In my case, it turned out to be an amplifier drawing too much power. Fortunately, a software update fixed the problem completely, and my battery has been great ever since.