  We consider a broad class of massive four dimensional effective theories
describing an infinite tower of charged massive spin 1 states, interacting with
massless spin 1 and spin 0. The spectrum is chosen to be the same as that
appears in the Kaluza-Klein theory reduction of 5d Yang-Mills to ensure the
absence of any spurious poles in a possible double copy formulation. The
effective theories are characterized by multiple different couplings between
different fields which are unconstrained by symmetries and low energy criteria.
Remarkably, by demanding that the scattering amplitudes preserve
colour-kinematics duality for different scattering processes, required for the
existence of a massive double copy, we find that our 4d Lagrangian is fixed
uniquely to the Kaluza-Klein compactification of 5d Yang-Mills theory together
with its known double copy consistent higher derivative operators.
