Yeah, as much as I hate to say it, a limited slip sounds like the best option.
Although, with the cost of buying one and then installing it, you might really see if you can step up to a selectable locker or something.
I kno wthat even the factory trac-loc in my TJ did better than a similar open diff'd vehicle in the snow. Likewise the trac-loc in my old Ram worked pretty well in the Dana 60.
But lockers are far above and beyond a LSD..
They are right, a lcoker isn't going to be all that un-locked around corners in something like a Jeep. And its going to slide around on snow and ice just as much.
Having driven a YJ before in the snow with a lock rite and 31x10.5 BFG ATs, it wasn't that bad. Having a tire thats good in the snow helps a lot. But you have to understand that it will change the characteristics a lot.
Really, its your call. A limited slip will help a bit off road, and be a little more slidey on-road. A locker or welded rear will help a ton off road and be a god bit more slidey on road.
Lockers/spool will defiantely wear the tires more.