We consider fourth order singularly perturbed boundary value problems with two small parameters, and the approximation of their solution by the hp version of the finite element method on the spectral boundary layer mesh from Melenk et al. We use a mixed formulation requiring only C0 basis functions in two-dimensional smooth domains. Under the assumption of analytic data, we show that the method converges uniformly, with respect to both singular perturbation parameters, at an exponential rate when the error is measured in the energy norm. Our theoretical findings are illustrated through numerical examples, including results using a stronger (balanced) norm.