An electric winch is not a great choice for winching logs for anything but very occasional use. They are very slow, and the duty cycle on most of the inexpensive ones (and some of the not-so-inexpensive ones) is low enough as to make it impractical for logging use. I own the HF Badlands 3500# winch Which I have mounted on an antique UTV. From the owner's manual: "Duty Cycle Rating: 5% (45 sec at Max Rated Load; 14 min, 15 sec Rest)". Pulling in the full 50' at the full rated load would take hours. You don't necessarily pull every log at the full rated load, but logging tends to be a long, continuous pull. These winches are generally better for short bursts: raising and lowering a snow plow, or getting unstuck from a mud hole (where you generally only need to move a few feet to get yourself out of trouble).
Add to that issue the fact that your tractor battery can't take the deep cycling, and that your tractor alternator is woefully undersized to recharge that battery in a reasonable amount of time. The solution just isn't practical for anything other than very light logs or moving heavier logs out of the middle of your trail (as opposed to dragging them 50 feet from off the trail).
If you do go this way, a skidding cone will significantly reduce the drag from the log skidding across the ground, reducing the load on your battery and increasing the distance you can pull without stopping to let the winch cool. Even better is a logging arch, particularly one designed to be pulled in at the end of a cable (in addition to being towed by your tractor). IMO, neither of those comes close to what a real PTO-driven logging winch will do for you. Once you have used one of those, you'll wonder how you ever got along without one.
