My thoughts
A deep cycle battery would not be needed. Being an electrical engineer, trying to figure out how to limit the current flowing to keep the battery charged while it is being used.. might blow the fuse as it would try to pull a lot of current during a heavy pull but not done it before to know. If you are going to use the truck battery, will need big cables to flow the power through them. Look at a forklift battery disconnect type device at the truck & trailer so you can plug it in when needed. This keeps the cables out of sight when not needed. Unless you plan on using the winch when the trailer is not connected to the truck, I would not keep a battery on the trailer, spend the money on the cables. They won't go bad over time like a battery does.
2013 D/A 2500HD Crew Cab<> B & W Rollover Ball, Bed cover, def deleted