I believe that most iterated torus links cannot be changed non-trivially by a Conway mutation, as follows. If you look at the JSJ decomposition of the double-branched cover, then each satellite torus that appears in the iterated torus construction appears. Thus, you cannot have an essential torus that intersects these satellite tori non-trivially. If you look at the double-branched cover of a Conway mutation sphere, you get a torus; if the mutation is non-trivial, then the torus must be essential, and so it lives in one of the pieces of this JSJ decomposition. Then you analyze the purported essential torus a little more (essentially in the torus knot case) and see that it can't exist.

There's a bunch of details in fleshing out this argument. The JSJ decompositions of link complements are actually kind of intricate; I think this is the canonical reference:

JSJ-decompositions of knot and link complements in the 3-sphere https://arxiv.org/abs/math/0506523

But that paper doesn't mention mutation. I think this argument should be standard; is it written anywhere?

In case it matters, I'm interested in the case of links of algebraic singularities. The paper is pretty algebraic as a whole, and I'd really rather not take an extended detour into JSJ decompositions.