6 pin has a charge wire (technically aux so it could be whatever you fancy).
Get some 12 gauge wire, 20amp fuse, zip ties, and some wire connectors(one that fits on your battery bolt). Run it from battery + to the plug on the bumper. Tuck it inside the frame, and out of the way, and moving parts. Give slack where necessary for frame/engine movement. Keep away from exhaust. Use e-trailers wiring diagrams for the pin out. You may end up realizing that both your truck and existing trailers wiring isn't standard. Mine wasn't.
Tapping into a taillight wire is a recipe for a blown fuse.
One other thing, you may want to tap into an ignition switched battery + or use a relay so that you don't drain your battery if the trailer has issues. (a relay allows you to switch a large load by applying a small load to the other side of the relay. That way the small load doesn't blow your existing ignition switched circuit, but the trailer charge wire only runs when the ignition is on.)
7 pin adds reverse light.
Let me know if I can help more. I don't think it's a bad idea to have the 7 flat rv plug too, since it is becoming common. It sounds like you still are going to have to run a power wire though. Oh, when you put the wires into the itty bitty terminals in the connector, make sure no strands are astray. I had a pulsing brake issue when I didn't notice my turn signal had one strand out of the terminal and touching the brake terminal.