In the question, it is not assumed that $\rho$ necessarily comes from an automorphism of $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$ .

Why is the normal bundle $\rho$-invariant? Are you assuming that $\rho$ comes from an automorphism of $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$?

I also expect that such a K3 surface $S$ is not a generic one. I simply would like to know whether such a K3 surface exists, regardless of generic or not.

@Jason star, no other requirements are put on, hence you answered the question! Being excited by such a quick answer, I changed the question a bit, which I would like to know the answer originally.

Ah. Thanks for the reference!

@Ennio Mori cone, good points. But I am doubtful about "....defines a cubic equation in many variables, so there are integral solutions..." because generic cubic forms do not tend to have nontrivial integral solutions even if its rank is very large. This is a crucial difference between two-dimension and higher dimensions.