I don't have time to think carefully.  However, I answered this question for the sphere $G$-spectrum here:
https://mathoverflow.net/questions/36455/.
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.