Another vote for a front Eaton/Detroit True Trac. Compared to other multi-tasking locking diffs, it has fewer and stronger parts with no clutches to wear out. It should be fine in a super 60 and is oh-so transparent on snow and ice. The other issue is: a rear Dana 135 was not available on a 2015 F-550. A Dana 110 and 130 were options for that year. The 4.88 was standard but you could also S.O. that tweener: 4.30:1. So, if you truly do have a D-135 on a 2015, it was put in after the fact.
I'm with Charlie in the recovery succession. I don't think the O.P. is so very hard core, so a couple limited slips should give the best service until you need to deflate, and have a way to get the tires back up once you are out of the stick. Oh, and locking hubs are a good investment. The trick, as noted above, is to not get your axles twisted up and lift a wheel. Yes, I"ve had almost every locker and L.S. ever made ( not the Ox Locker) and don't think a full Detroit Locker would hold up with that much GVW. ARB's would work, but it's about double the price once you connect the plumbing, compressor and install. With my little 10.4K pound truck camper I'm running a 35 spline faux/super 60 in front with a True Trac (only the case and RGP are not Dana 70 spec) and a 35 spline Dana 80 in rear with a True Trac. I don't have the woes on ice and snow I had with full time, non-selectable locking diffs, or even my Dana 80 Power Loc, which had a stiff preload, with the attendant crossroad of oversteer under accel and the understeer on decel. I'm thinking the most important thing in all this is the moxey and off road chops of the operator. Here is a pic of a Dana 60 (or 70), 35 spine True Trac along with one I put in my '99 XJ's HP Dana 30 for contrast:
jefe