Trailer battery charging from Ford F250 (tow vehicle) How to wire?

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Trailer tow connector.
Brown wire with white tracer is voltage for park lamps, comes from trailer park lamp relay.
The yellow wire is voltage for left turn, comes from left turn trailer relay.
The dark green wire is voltage for right turn, comes from trailer right turn relay.
The white wire is ground, check voltage and ground. Use a test light. I don't know what testing you have done?
There are several diagrams for this system.
bbbind.com has free diagrams.

The fuses for the trailer tow adapter are in the normal fuse panels.

Engine Compartment fuse panel
101 - Is the battery charge circuit ( hot in run position )
20 - Left Stop / Turn fuse
19 - Right Stop / Turn
5 - Trailer tow reversing lamps & Trailer Tow parking lamps

Cab Fuse Panel
23 - fuse for the relay for the trailer tow battery charge circuit

All of these are in the owners manual

If you did not get an owners manual with your truck ( purchased used ), you can download one from Ford :
https://www.fleet.ford.com/maintenance/owners_manuals/default.asp

Most likely the brown wire from the trailer connector is connected to a constant battery voltage source, and it should be connected to the vehicle running light circuit, however someone might have wired the trailer connector up that way on purpose so that the trailer does not draw power from the vehicle running lights and possibly damage the vehicle running light circuit. (Dodge/Chrysler vehicles do have delicate lighting systems) Here are some images to help assist you, and the 4 way connector is the same as the 5 way connector just without the blue wire for the trailer brakes.

Might want to bench test the trailer tow battery charge relay.

You can also jumper this with a wire in the relay socket, to test the wiring to the trailer tow adapter. This is relay R05

That would be my first guess,they are not very exspensive,if this still does not help,run a large wire on the back of the alternator with the old one still connected,to the positive battery post.This will increase the voltage to the battery also.Put it on the large batt,connector on the alternator,and run it to the battery post+

If you have a stick shift, you rent a 2 wheel tow dolly and run the front tires onto it and strap it down. Put the truck in neutral and away you go.
If you have an Automatic you want all 4 wheels on a flatbed or tow trailer. You rent a car hauler trailer and piggyback the truck onto the trailer. A good rental will know the weight and rent you the right trailer.

If It Factory Ford Tow Package And They Have Not Changed Things There Is A Fuse In The Fuse Panel For The Trailer Lights Hope This Helps

Go to this web site: http://www.genuineservice.com/genuineservice/en/default?page=genuineService Pick owner guide on the left task bar. Input your vehicle info then pick get owner guide. Pick owner guide 5th printing and your fuse info begins on page 176

Battery junction box (BJB) fuse(s):

• 5 (30A) (trailer electric brake controller)
• 34 (25A) (trailer tow turn relays)
• 42 (30A) (trailer tow parking lamps)
• 44 (30A) (trailer tow battery charge relay switch side)
• 63 (15A) (trailer tow reversing lamps