I don't have time to think carefully. However, I answered this question for the sphere $G$-spectrum here: A heart for stable equivariant homotopy theory. I see no problem in generalizing that answer to any connective commutative ring $G$-spectrum $A$. Using cell $A$-modules, one can kill the higher homotopy groups of an $A$-module, etc. The Mackey functor $\underline{\pi}_0 A$ is a Green functor, so one can form module Mackey functors over it, and the heart should be the Eilenberg-MacLane $G$-spectra of such modules, with appropriate structure as $A$-module $G$-spectra.

I don't have time to think carefully. However, I answered this question for the sphere $G$-spectrum here: A heart for stable equivariant homotopy theory. I see no problem in generalizing that answer to any connective commutative ring $G$-spectrum $A$. Using cell $A$-modules, one can kill the higher homotopy groups of an $A$-module, etc. The Mackey functor $\underline{\pi}_0 A$ is a Green functor, so one can form module Mackey functors over it, and the heart should be the Eilenberg-MacLane $G$-spectra of such modules, with appropriate structure as $A$-module $G$-spectra.