We perform the analysis of the hp finite element approximation for the solution to singularly perturbed transmission problems, using Spectral Boundary Layer Meshes. In [12] it was shown that this method yields robust exponential convergence, as the degree p of the approximating polynomials is increased, when the error is measured in the energy norm associated with the boundary value problem. In the present article we sharpen the result by showing that the Zip-Finite Element Method (FEM) on Spectral Boundary Layer Meshes leads to robust exponential convergence in a stronger, more balanced norm. Several numerical results illustrating and extending the theory are also presented.