My winch is speced with a 4.1Hp motor
1HP = 745.7 Watts
4.1HP = 3057.4 Watts
3057.4 Watts = 244.6 Amps @ 12.5V
When the output of the alternator is overwhelmed, the battery acts as a reserve. This is why the lights dim and it's known as voltage sag. As the motor is running the current demand continues. Which causes the voltage to sag further. Since you cannot generate more average power than the charging system has available, that 4.1 HP motor is going to produce less power (2V sag = 30% loss of power).
What can you do about it?
Well, you could upgrade the charging system with a new alternator. The Jeep will need 60-80 amps to operate so the alternator will need to be hot rated @ 325 amps. I don't know of an alternator capable of this so you could get a custom bracket made and add a 250 amp alternator with another battery dedicated to the winch. I don't know how to fit it all, but that's the deal.
OR you could simply not worry about it. How often will you use your winch? Even when you do, how long will it operate? I think alot of peeps use their winches without problems. I think that it would be fair to say that when they are used, they are used as minimally as humanly possible.
If you're battery is no longer in good shape, a fresh one will help alot!
An 8000lb winch will be fine and would do the same work as a higher-rated winch connected to the same charging system.
The universe is ruled by the laws of physics not by the laws of man (or marketing departments of winch companies)

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2006 CRD
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RL rear bumper, ARB front bumper, Mile Marker winch, Power Tank, Skids by Mopar & Skid Row, RL Super Skink Sliders, 235/85R16 MTR, Air Flow Snorkel, DTT, 4.10 gears, Frankenlift II, Katskin leather, JL Audio stereo, Rosen video.