Winching takes huge power. A 3000lb pull could draw 200 amps (varies with the effective drum size).
If the winch is going to work, the juice has to come from somewhere. The only place that could deliver that much is the battery.
Batteries are odd things - they will never deliver all the juice they are said to hold and how much they will deliver depends on how fast they are being asked to deliver (other factors like age and temperature also apply).
The battery can be supplemented by the charging system. The generator output of 850w is the MAXIMUM it will supply. At engine idle speed, the net output is likely to be very low.
So, when winching, always have the engine running above idle speed. Winch in short bursts so that the battery can be stopped up in between.
If you use your winch often, consider wiring the winch to a second battery which is wires with a split voltage sensing charging system.