  We investigate the realisability of the Casson-Sullivan invariant for
homeomorphisms of smooth $4$-manifolds, which is the obstruction to a
homeomorphism being stably pseudo-isotopic to a diffeomorphism, valued in the
third cohomology of the source manifold with $\mathbb{Z}/2$-coefficients. We
prove that for all orientable pairs of homeomorphic, smooth $4$-manifolds this
invariant can be realised fully after stabilising with a single $S^2\times
S^2$. As an application, we obtain that topologically isotopic surfaces in a
smooth, simply-connected $4$-manifold become smoothly isotopic after sufficient
external stabilisations. We further demonstrate cases where this invariant can
be realised fully without stabilisation for self-homeomorphisms, which includes
for manifolds with finite cyclic fundamental group. This method allows us to
produce many examples of homeomorphisms which are not stably pseudo-isotopic to
any diffeomorphism but are homotopic to the identity. Finally, we reinterpret
these results in terms of finding examples of smooth structures on
$4$-manifolds which are diffeomorphic but not stably pseudo-isotopic.
