Was daytime running lights listed on your invoice when you bought the truck? Its rather foolish to assume that what is standard on one vehicle would automatically be included on another.
I have a simple fix. Turn your lights to AUTO and place a small piece of electrical tape over the light sensor. Lights automatically turn on once the truck has started. Then through your EVIC you can adjust length of delay after your truck shuts off.
Another option is to simply turn the lights on every time you get in the truck.
Either way, you get lights turned on while driving and you power the running light circuit, therefore providing power for your reverse camera on your trailer.