You currently have just the black 7-pin socket on your car which is why the trailer battery isn't charging (the revese light won't work either.) I have both types of socket on my car- 13 and 7-pin. Means I can tow anything without messing about with adapters which don't always work. The old system of twin black and grey 7-pin plugs (which you'd need for the trailer battery) takes up the same amount of space, and adding the 13-pin while retaining the 7-pin gives more flexibility.
Wiring the lights is easy enough on these as they just splice into the bullet connectors behind the lights. All the road lights for both plugs can be connected up this way. You can now legally tow any trailer regardless of which connector it uses.
For the auxiliary power supply, there may already be a purple wire bundled up in there which is a permanent fused supply. Use this to power pins 9 and 10 via a voltage- sensing relay. Mostly these expect you to be using the old grey plug so the "pin 4" output goes to pin 9 and "pin 6" goes to pin 10.