i'm not so sure it's as simple as you're making it. in the initial state all cables are tensioned against the crate, so there's a unitform pressure between the crate and the ground. a cable that goes into compression doesn't "just" disappear, it reacts load untill the prelod is exceeded (and then it becomes slack). if you remove the slack cable from the initial problem, then you've changed the initial situation.
i think you need to say the cable becomes slack when x% of the loadcase is applied, and 1-x% is rected by the remaining three cables. maybe easier to see by superposition ...
1) initially all cables have tension, Ti,
2) apply x% of the load, reacted by four cables, get cable loads (one should be -Ti),
3) apply 1-x% of the load reacted by the other three cables,
4) sum the cable loads (one will be zero).
you can calculate x by applying 100% load seeing the compression load in the cable, C, then x = Ti/(Ti-C) ... C being -ve.
instead of a myriad of cases, can you examine the corners of the envelop, 8 cases ? how much precision do you need ?
another day in paradise, or is paradise one day closer ?