  The exploration of the massive double copy is still in its infancy, and only
a few examples in limited contexts are known. In this paper, we analyze the
newly discovered double copy for topologically massive theories beyond
tree-level amplitudes in the high-energy (eikonal) limit. We find that contrary
to the simple double copy relation that occurs in the massless $d\geq4$ case,
the massive double copy requires information outside the eikonal limit to give
correct results. However, a simple double copy relation arises for the phase
shift. Additionally, we relate the eikonal amplitudes to the corresponding
shockwave backgrounds and find a classical double copy relation that is
manifest only for the appropriate choice of boundary conditions. This analysis
hints towards the existence of a topologically massive double copy at all loop
orders.
