  In previous papers of this series we analysed the reduced phase space
approach to perturbations of Einstein-Maxwell theory to second order around
spherically symmetric backgrounds in the Gullstrand Painlev\'e Gauge and
confirmed consistency with previous approaches. In this paper we generalize
this result and show that the analysis can be performed in gauges for the
background variables compatible with the Gullstrand Painlev\'e gauge. We obtain
the same structure for the reduced Hamiltonian that contains the well known
Regge-Wheeler and Zerilli potentials. Possible applications of this
generalization are discussed.
