Not arguing about putting a bigger battery in or anything, just clarifying what this does. A "bigger" battery, in regards to cold cranking amps (CCA) that everybody looks at, does not make the winch work better or have more power. You need to also look at the reserve capacity, and an increased reserve capacity and higher CCA will simply allow you to run the winch longer before draining the battery. Even with the engine running and alternator working the winch can easily draw more power than what the alternator can produce and thus is pulling reserve capacity from the battery and will drain it. A Warn 12,000 winch is rated to pull up to 440 amp at maximum load which is at least double the maximum output the alternator is capable of producing. Keep in mind the alternator output is dependent on how high you are revving the engine. Even a 220 amp alternator is probably only producing 100 amps at idle.
Overall, my point is that upgrading from the factory battery to a little bit bigger battery has very minimal advantages in regards to winching. And that advantage is maybe an extra 30 seconds of running the winch at a very high load before draining the battery.