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Scharlemann and Thompson proved that one can isotope/slide a tunnel to be disjoint from a minimal genus Seifert surface (which was what I had in mind in my comment above). Their argument does not depend on it being a knot complement. I also found a citation for a paper of Sakuma which claims to prove the result for punctured torus bundles, so it might have the result you're looking for (although I haven't looked at the article yet).