When you said you had battery checked, was it load tested or just voltage measured? If you take it to Autozone or the like they will hook up a load tester that tells the real condition of the battery capacity, not just voltage. The winch isn't underpowered, it might be drawing too much power for the capacity of the battery. How is the battery charged? Perhaps the charging system isn't fully charging battery.
Bob & Donna
'98 Gulf Stream Sun Voyager DP being pushed by a '00 Beetle TDI