The double copy procedure relates gauge and gravity theories through color-kinematics replacements, and holds for both scattering amplitudes and in classical contexts. Here, we extend this correspondence to exact classical solutions of maximally symmetric curved spacetimes. We consider asymptotically (A)dS spacetimes in Kerr-Schild form and construct the corresponding single and zeroth copies. We focus on understanding how to extract the Yang-Mils and biadjoint scalar field equations from Einstein equations. We also explore the double copy in (2+1)-dimensions by examining the cases of the BTZ black hole and the double copy of a gauge theory point charge. The latter gravitational solution is the first case of the Kerr-Schild double copy involving a dilaton. Furthermore, it has been shown that there is a web of theories whose scattering amplitudes are related through operations that exchange color and kinematic factors. We generalize and extend this procedure by showing that the classical perturbative double copy of pion radiation corresponds to Special Galileon radiation. We also construct the single copy by mapping the biadjoint scalar radiation to the non-linear sigma model radiation through generalized color-kinematics replacements. Afterwards, we compute the higher derivative amplitudes arising from shift symmetric-invariant actions for both the non-linear sigma model and the Special Galileon symmetries, and provide explicit expressions for their Lagrangians. We find that, beyond leading order, the equivalence between shift symmetries, enhanced soft limits, and compatibility with the double copy procedure breaks down.