My greatest concern would be if you are on that 20% slope and there is wet clay sitting on rock. I'd leave FEL bucket really low ready to dig in. If there's a toothbar, I'd have that on. I also would go very slowly there, because coming to a quick stop with much speed under those conditions could result in fishtailing. Fishtailing on that slope could put the trailer into a roll, torquing the tongue of the trailer and the coupling to the hitch and all kinds of not pretty things could result.
To what length will these logs be cut prior to loading the tractor? For long logs, there are 3ph devices especially designed to grapple and pull in a straight long line. Cutting long lengths, dragging them out and then recutting would be a PITR and I would surely think you are cutting into fireplace length prior to loading tractor. That probably being the case, have you thought about a carryall on the 3ph? This would avoid that hinge at the hitch and fishtail problem. You could even put firewood on a pallet, strap it, then back the carryall forks into the pallet. Similarly, putting forks on the front of the FEL would place the cargo in front of you on that slope. A pallet of firewood on forks out in front of the FEL would likely put the CG too far out in front and tip you forward unless you kept bh on as counterweight, but that makes the entire assembly so long and narrow and bouncy that it could go into a side skid on that wet slippery slope and end up in a roll. Putting forks on the FEL bucket with cross members extending only a short distance beyond the FEL front lip, then loading the firewood into the bucket and just past the front of the bucket and strapping it would keep the CG closer to the center of the tractor.
I would not use the trailer on that slope in any moist conditions. I'd just load the FEL or buy a carryall (they only cost ($99-$149) and get some cheap used pallets. You'd stack the firewood one single time on the pallet and unstack as needed for the fireplace. When you get to the house or barn, just drop the pallet with wood still on it and can easily move loaded pallets around as desired later if needed. You don't unload anything off a pallet until you are ready to take a few armloads into the house.
Actually, instead of strapping the wood on the pallet, you could buy a big roll of shrinkwrap and secure it really well. If you continue the shrinkwrap over the top of the stack of each pallet, you could even waterproof the entire load. I guess this would be best for trees that are dead-standing and have already seasoned/dried out. Wrapping green wood completely wouldn't let it dry very well and it'd likely get mold, fungus, and other decomposers started working a lot faster, so for that I'd leave the top not covered.