Can you set your bales in the fall and then unroll them by hand (push them down the hill)?
For the last 6 years I have fed hay by bale grazing on the side of the hill in KY. I pick the hill with little to no grass or one I have recently cleared trees from in the past year. I set bales in a grid pattern ahead of time, in the fall - when it is dry and the ground is solid. On the steeps hills it is necessary to use blocks on the down hill side of the bales to keep them from toppling and rolling down the hill. This winter I didn't even need to start the tractor to move hay or drive in the mud. I only have an old 2 WD tractor. I move an electric fence and bale ring every couple days, as the cows finish the bales. I use the cheap bale rings, because they are easier to move (lighter). I don't have a huge pile of manure at the end of winter to deal with, it is all out there on the hill.
I will update my post on clearing land from last year and show what it looked like this spring and summer. I thought it turned out well.
When feeding on existing grass, the cows can rut things up pretty good. I have noticed that the first year after feeding hay in an area the grass is pretty stunted. Two years after it starts looking better. I can still see the areas I fed hay on 3-4 years later, because the grass is superior to the areas around it. I sure would not go back to moving hay around in the winter. What a mess!
Your mileage may vary.