I played around doing some mild 4x4 / off road things for a few years and have a little experience with winches.....
I got a good used Warn M-8000 off CL for about 400. I mounted it to a home made portable cradle (piece of C channel, 2" tube stock & 3 huge bolts) that plugs into the front receiver hitch on my S-10 ZR2. I never had to work it hard when wheeling but it never let me down either.
I got 2 different log loads delivered to the house. A tri-axe load of 16-20 footers and a tractor trailer load of similar sized logs. The Warn and S-10 did a nice job of dragging the logs off the top of the pile for easier bucking. When getting low in the piles, instead of cutting in the dirt, I would chain a log up to the cable and run it through a pully block that was hanging about 10 feet up in a 12" pine tree. Poor man's crane hoist system. Bucking at waist height was nice.
A series wound winch motor like this can draw upward of 400 + amps when working hard. The stock battery and 100 amp alternator were becoming taxed when at idle. Mostly because the alternator only puts out about 40 amps at idle. I upgraded to an AD-244 alternator. It bolted right up and plugged right it to the 4.3 L motor. The case is only barely bigger than the stock one, but the AD-244 puts out 100 amps at idle, does a max of 145 and cools itself better. The AD-244 is what comes stock on a full size GM truck with the HD electrical system. My Duramax has one as stock equipment. The only way to go bigger from the factory is to order dual alternators, but you get dual crappy 100's, which could be swapped out to 145's, etc.
We have an '06 Jeep Unlimited that is lifted 4.5" and running on 33's. I put a Smittybilt XRC-8 on it. I got that thing stupid cheap from
www.winchdepot.com for 269 shipped. It's a good winch and has faster line speed than the Warn. We used the Jeep to drag a few logs out of the woods here and there and usually I have to strap the rear end to another tree or the winch will just drag the Jeep around like a rag doll.
I have a 16 foot 14 K GVWR trailer that I originally bought to haul the Jeep to wheeling events with. I've hauled more wood, mulch & stone on it though, LOL. I have not gotten around to mounting a winch on my trailer yet, so I will be interested in seeing how your project turns out.
I would think something in the 8-10K range would be a minimum starting point. You can always rig up a snatch block for 2x the power but that will cost you by being 2x slower line speed.
I would not want to use synthetic line for wood work. Too much potential for abrasion damage and that stuff is costly, but wire rope gets heavy and needs constant attention so you don't "birds nest" it on the winch drum.