OK, I looked, and there doesn't seem to be Upfitter wiring for the '05.
There is in newer models.
I don't know the wiring of the t-connector, but the way to make a safe connection between the truck turn light circuit and the trailer connectors is with a relay.
Splicing another bulb directly into the circuit can damage the driver in the Front Control Module for the rear turn signal.
Unlike the trailer tow turn signals, they do not use a relay connected to a fuse.
Instead, it is driven by electronics, and cannot take the load that a relay and fuse can.
You can use the truck circuit to control the relay for the trailer lights.

 

I know this is an old post but wanted to share my experience. I had the same issue. Routing 2 wires from the truck signal light into a trailer relay with one wire coming out to the trailer plug in does work. I've done it and it lasted for many years. Years later same thing happened but truck left signal light didn't work. Went to the dealer, lady said have to diagnose the problem, etc etc $125 or so an hour. I asked if there was a problem with modules, she said they replace tons of them, $1500 or so. Went to another dealer in the country. Told guy first dealer said they could reprogram and it would solve the problem but they didn't have time to do it right away so I came to them. Lady refers me to some young guy from the back. He said he never heard of that but would try. Came back later all excited, said it worked. Charged me $80. Apparently it can only be done so many times. It happened to me twice and both times I had either lent my truck to someone and they hooked it to another trailer or a trailer plug in wire got grounded or powered by mistake. Don't remember which. No more hooking up other trailers to my truck.