All great advice. Might try heat cool cycles, heat to red 400-550° then might try candle wax and let cool (map gas may not be hot enough). I weigh over 300lbs and bounced on my 1" braker bar with a 3 5/8" nut. That didn't work as well as I wanted so I gathered up a 15' cheater bar and bounced on it with everything I had until the nut was torqued to ~ tic tic pounds.😁 ~1400lbs - 1800lbs total guess. A good USA made 1' breaker bar should take the abuse.

Yes, 18 volt Millwaukee brushless gun 3/4' can do wonders. If that does not do it alone, try the heat method. I wonder if the nut is LEFT hand threaded? Last resort would be to split that nut with a die grinder or such. It would be nice if someone familiar with that BIG nut and winch could chime in. Nothing is impossible, it might just take a little longer....JM

3000 ft lb Torque Multiplier, it will give sooner or later. lol

Yes, 18 volt Millwaukee brushless gun 3/4' can do wonders. If that does not do it alone, try the heat method. I wonder if the nut is LEFT hand threaded? Last resort would be to split that nut with a die grinder or such. It would be nice if someone familiar with that BIG nut and winch could chime in. Nothing is impossible, it might just take a little longer....JM

You know, I was thinking that too about a left hand thread after I posted .  Clean the paint off and take a close look at the thread  and you should be able to see if it's starting counter clockwise.
 

might be some helpful information in this video

It’s not a left hand thread, just a good size fine thread nut with a lot of tension on it, that rod is a major part of the structure on the winch, holding the drum and the cast steel housing assembly together, they are tight, I have a 2800 ft/lb spline drive impact wrench, usually have to stand there and let it whack away for a couple minutes, uses a lot of air but they always come.

Hi Neil, when an apprentice we were told that 1 inch of diameter will grow 4 thou with heat (can't remember temp range for it), first heat will shrink the most and the second a little and the third generally shrink is minimal or none,
Was told by a boilermaker that heat the pin not the bush/boss/clevis so it can grow to crush tighter in the tolerance, this will transfer heat out to the bush, then cool pin only slowly to allow it to shrink pin and heat to remain in bush/boss/clevis to be expanded slightly to let go of pin, one day we heated a steering cylinder pin red hot at ends, cooled with trickle from hose and left jack under the pin to push out, we went for smoko and heard a bang, pin came out in hurry dented cab floor underside, pin out with heat and cool only(no hammer work).

True on the air, used a 1" impact to remove track bolts with two cylinder shop compressor, just didnt have the volume, got the Joy out with the D315 at 210 cfm and wulla, spun em right off.