Have you considered putting a battery on the trailer instead of trying to run everything off the rig? Usually trailer winches are only used for short pulls, so a fully charged battery has more then enough juice to do the job. If you still want extra power you can get away with running 8 ga wire from the truck battery to the trailer battery to keep it charged.
Otherwise, if I'm doing the load equation properly it looks like 2ga for the 20+ foot run figuring the 300 amp draw. Problem is, I'm not sure there is a commercial connector that works with that large of wire - most of what I see is for the 4-8ga size stuff.