There is indeed a 12 volt BATTERY pin on the connector, however it may be "HOT" only when the towing vehicle is running... While I forget which pin it is, I do know it is next door to the "Tail Light" pin. And that it is not always connected.
Try connecting, via a fuse the + pin to the two pins on either side, if lights come on it's 12 volt battery hot. The pins (Tail Lights and Battery) are opposite the the keyway, Right Turn is on the other side of Battery.. So you can connect battery to either one and confirm.
7 Pin Connector
May be of help