Harrison You have a few ways that you could approach solving this - give that you have attached your skinCluster directly to your tissue, you could paint the weights of the "zEmbedder" to a value of "0.0" in the areas where you would like the mesh to follow your skinCluster.
Alternatively, you could have a seperate mesh for your tissue and you skinCluster and blend the results together using a blendshape node.
I hope this helps!
-jj