Ask an Electrician
First off let me state that there are no light fixtures that are non LED compatible. An LED lightbulb will fit in any and all normal light sockets. The only difference in LED bulbs is if they are made dimmable or not.
As far as your problem with one socket burning out bulbs is due to a heat issue. The heat issue can be caused by loose connections or worn parts that don’t fit correctly. most sockets can be replaced with parts from a quality hardware store. Just make sure that when you do any repairs that you tighten and fit all parts so this problem is everted in the future.
Thanks guys. With all of it being kind of "non-standard" I didn't know if I would get a licensed electrician to even want to touch it. I guess I have to ask first! I'll start out next spring by getting someone to look at it. Ideally, I would be able to install a wireless remote to the existing winch. I'm not fond of the current "cable control" and spring switch it uses now. Thanks!
The transformer puts out 12V AC, not DC. You would need to add a rectifier and filter. Plus, if you note, the winch is rated over 3000W peak, and if you price the 5kVA control transformer you would need, it costs more than the winch.
A good electrician could figure out your issue for a lot less. If the motor has power but is not turning, you probably lost a starting capacitor, will cost you $20 to fix. Plus the Electrician's time, of course.