I haven't been able to find an answer for this so here it goes!:

I've got these two clevis connectors mated in a slot. Imagine that there is a spring going off axially from each clevis. When you rotate the main part one clevis will become unloaded where the other becomes loaded by hitting the end of the slot. Is there a way to create a mate that models a spring force? Minimizes the distance between two parts, while maintaining the constraints of all the other mates?

Calculate the distance and then try using limit distance mate.