Can we protect PRESENT from the Statistical Saturation Attack if we change the permutation layer to a random permutation layer?

Question

This is continuation-question based on: If PRESENT had different permutations s would that protect it against Statistical Saturation Attack? Can we protect PRESENT from Statistical Saturation Attacks by using:

1. unique perms for each round or just some rounds? (In this case, how many might some have to be?)
2. random, key-dependent (say), perms for some or all rounds?

Moreover, if these ideas can indeed protect PRESENT against SS attacks:

3. Could we use fewer than 31 rounds without causing vulnerabilities to other attacks?

See also What affect does the permutation layer have on the avalanche effect?