I just had my 2014 CRV set up for towing with a Blue Ox system with a Patriot brake, which is like your Brake Buddy. I had the tech install the switch, and when it is thrown, the power to both outlets is killed. Grrr. He didn't bother checking, and I didn't find out until I brought it home and went through a hook up dry run. We had to install a separate outlet that is direct wired, and not switched.
2003 Rexhall Rexair, 3550 Ford V10 with Banks
2014 Honda CRV-AWD via Blue Ox
Travel with SWMBO and Golden Doodle co pilot