We tow our Terrain (same as the Equinox...just different body style) 4-down, but have the US Gear Unified Brake system for the car when towing so no additional battery drain as you have with the Patriot system. As mentioned, you just need to provide a different connection for the 12v. HOWEVER, unless you have a auxiliary 12v line coming from the motor home's battery(s) to the toad battery, the Patriot will drain your toad's battery.
If you don't pull fuse 32, you will also have a dead battery once you're at your destination.
I made and installed a switch to toggle fuse 32 off and on so that I wouldn't have to go through the uncomfortable position to pull/install the fuse.
Ron