Joe,
I actually have the Blue Ox Patriot braking system vs. the Brake Buddy, I just used the brake buddy as a generic reference like saying Zerox vs. copier. That being said I assume the Brake Buddy operates like the Blue Ox and it does use the vehicle power to operate the system. I don't have any electrical specifications on my unit but I can't imagine the system requiring much power during normal highway driving and I usually keep mine on the lowest sensitivity setting.
According to the FMCA article there are other systems still powered on while towing, depending on which fuse you pull. F32 disables the electric power steering and F16 disables the ABS system, if I'm reading these documents correctly. I wasn't aware of this and wondered why my battery kept dying even with shutting off everything I could.