Not knowing which winch it really is, to look at the wiring schematic, I think the wire you're talking about is the ground wire for the solenoids. Is it about a 14 or 12ga?
If so, that's what it's for and it can be grounded to any good ground point in your engine bay.

True, but wire has resistance regardless of the duty cycle at which you're operating. Too small a wire and your winch isn't going to work to its full potential above a certain amp draw, even if you're only only running it for one minute.