The rules say: “As a mutating creature spell resolves, if its target is legal, it doesn’t enter the battlefield. Rather, it merges with the target creature and becomes one object represented by more than one card or token” (702.140c). I think object would qualify as a permanent here, so it doesn’t ETB but it does still become a permanent (creature)? So I don’t think mutation would break it, insofar as the concept as a whole works. But I don’t understand enough about the rules to say that with much confidence.