Here are some options as I see your situation:
1) Keep looking for someone who either has or will make tracks for the Case loader. Personally, I've never seen a large loader with tracks, but that doesn't mean they aren't out there. Custom designed ones are going to be cost prohibitive, no airport cares about the grass that much. The other thing you may want to check on with tracks is that although they decrease contact pressure, they can really tear up sod in a turn. I've never seen the articulated tracked ag tractors used on a sod surface so I don't know what they do, but the ones that steer by only driving one side or by counter-rotating the treads can really tear up grass quick.
2) Abandon the Case loader all together and look for track mounted equipment or the ag tractors w/tracks to see if they could be adapted. You may have some luck looking at dozers, the CAT Challenger, the JD tracked ag tractor, or the other ag tractors to which tracks can be added. Dozers might be your best bet as far as having all the capabilities out of the box, but I don't know if rubber tracks are available in the size you'll be looking at. Reference the tread issue I brought up before, my guess is the operators are going to be looking at getting the job done quickly, they're not going to get too concerned about the grass under them so you'll see lots of real tight turns. This too is a $$$ option and I can't believe they like grass that much.
3) You could look at options as far as different wheel/tire combinations for the Case loader. A set of wheels to use a floater style tire, a simply a set for wider tires, or even duals would decrease your ground pressure. Even if you had to have some custom wheels made, I would guess they would be considerably cheaper than options 1 or 2. They would also be interchangable with the rest of the fleet. Traction may end up being an issue with this option, so you'd want to check into that.
Hope this helps. Definitely an interesting problem to solve.