If your Tundra is rated at 10K that’s 10K on a road at road speeds. As in my Tundra pulling my 9500lb 30’ camper from the middle of NC to Pigeon Forge in the Tennessee mountains (and there are some 8% grades in that route, too). It can pull a good bit more 600’ slowly. The limiting factors on the Tundra would be tongue weight (easy to check, just hook it up and if it sets the axle on the bump stops it won’t work) and ability to stop (if the trailer doesn’t have brakes, but the brakes on just the Tundra are a lot more than what you have on the skid steer).
If I had to move a 15K trailer 600’ and had to choose between a skid steer that has no real brakes and my 10k rated Tundra, that would be the Tundra.
Whatever you do, good luck with it and stay safe.