Some care is needed in using the term, "locker". Most of the rigs above have limited slip diffs, traction control (where brake pressure is automatically put to the spinning wheel engaging the one not spinning) or a hybrid version. The good news is L.S. diffs have come a long way in the past decade as has the capability of small SUV's. The producer of the the short vid going up Hell's Gate, which I've done in my CJ-8, would have done himself and the people in the demo car a big favor by lowering the pressure a bit to obtain more traction and a smoother ride. The vid does NOT show how really steep that sucker is.
I rented a small Japanese SUV 20 years ago in Denver for a conference in the Rockies. It had all wheel drive, so I figured I'd take it for a spin when the business day was over. So, up some canyon and then a side tributary i go with the auto trans in 1st gear and it climbed pretty good for a while. Within a minute or two the engine kept revving but no forward motion, and limited wheel travel, just a strange scorched smell. Scratch that one off the list.
Jeanie drives a 2011 Grande Cherokee Limited with all the ghee-ghaz, with low range and the best traction system Jeep had to offer (i think Quadradrive II) and I find it rather jerky. You must have wheel spin first before the limited slip part catches up. First time we used it in the snow, I backed out of my garage and the traction malfunction light came on and the machine was marooned in neutral for the day. I'm not so thrilled with the 5-speed automatic either. My first choice considering the O.P.'s constraints would be a 1992-2001 Jeep not-so-grand Cherokee Classic. I've owned 3 of them and still have one floating around the property. Of the 13, 4WD's I've owned, the XJ is hands down the best snow car ever with True trac front diff and trashlok rear. One car guy/writer's editorial on the 20 best cars ever built, and that includes a lot, includes the Jeep XJ; the only 4WD on the list. There is a reason they built these from 1984 to 2001.