We present a theoretical description of the effect of polyvalent ions on the interaction between titratable macroions. The model system consists of two point-like macroions with dissociable sites, immersed in an asymmetric ionic mixture of monovalent and polyvalent salts. We formulate a dressed ion strong coupling theory, based on the decomposition of the asymmetric ionic mixture into a weakly electrostatically coupled monovalent salt and into polyvalent ions that are strongly electrostatically coupled to the titratable macro-ions. The charge of the macroions is not considered as fixed, but is allowed to respond to local bathing solution parameters (electrostatic potential, pH of the solution, and salt concentration) through a simple charge regulation model. The approach presented, yielding an effective polyvalent-ion mediated interaction between charge-regulated macroions at various solution conditions, describes the strong coupling equivalent of the Kirkwood-Schumaker interaction.