You can find it in the paper: "Unknotting number, genus, and companion tori" by Scharlemann and Thompson MR0929535

They use sutured manifold theory to show that if a crossing change reduces the genus of a satellite knot by at least 2 then the loop along which Dehn surgery is done to effect the crossing change can be isotoped to be disjoint from each essential torus.

Off the top of my head, I don't see a way to convert this into a statement about diagrams.

The reference is given in the paper you cite: M. Eudave-Mun ̃oz, Primeness and sums of tangles, Trans. Am. Math. Soc. 306, 773-790 (1988)

The arguments are purely combinatorial, but there should be a simplification using sutured manifold theory along the lines of Scharlemann and Thompson's paper "Unknotting number, genus, and companion tori" by Scharlemann and Thompson MR0929535

Off the top of my head, I don't see a way to convert either type of argument into a something about diagrams.