  We determine the nonlinear stability of shock-fronted travelling waves
arising in a reaction-nonlinear diffusion PDE, subject to a fourth-order
spatial derivative term multiplied by a small parameter $\varepsilon$ that
models {\it nonlocal regularization}. Motivated by the authors' recent
stability analysis of shock-fronted travelling waves under viscous relaxation,
our numerical analysis is guided by the observation that there is a fast-slow
decomposition of the associated eigenvalue problem for the linearised operator.
In particular, we observe an astonishing reduction of the complex
four-dimensional eigenvalue problem into a {\it real} one-dimensional problem
defined along the slow manifolds; i.e. slow eigenvalues defined near the tails
of the shock-fronted wave for $\varepsilon = 0$ govern the point spectrum of
the linearised operator when $0 < \varepsilon \ll 1$.
