Yes, it's normal for the brake controller to be wired directly to the battery (through a fuse, of course). The idea is that under no circumstances do you want to be able for it to lose power when in use - for example, if your gas pedal sticks, and you kill the ignition switch to regain control, you don't want the brake controller going dead (assuming your steering wheel doesn't lock.)
I simply remove my P3 and put it into its case, and put that in the glove box when I am not towing. It takes 30 seconds to do.
Tangent: I *also* remove my hitch from the receiver, put the cap on the greasy ball, and stow that - to keep the ball well-greased, and to protect both my and other people's shins!