Although I agree with Peter's comments, I believe I can add a few helpful comments of my own. First for $G$ finite, every rational Mackey functor is both injective and projective, so chain complexes are weakly equivalent to their homology. One can find this statement in appendix A of Greenlees-May 'Generalized Tate Cohomology.'

Given a representation sphere $S^V$ and a graded Mackey functor $M$ with associated Eilenberg-MacLane spectrum $HM$ we have an associated $G$-spectrum $S^V\wedge HM$ which represents the integer graded Bredon cohomology theory $$ X\mapsto H^{*-V}_{(-)}(X;M ).$$ Because rational equivariant cohomology theories are ordinary there is an equivalence $$S^V\wedge HM\simeq H(\pi_*^{(-)}(S^V\wedge HM)).$$ Hence the representation sphere takes a graded Mackey functor to another graded Mackey functor, namely the Bredon homology of $S^V$ with coefficients in $M$.

A similar argument shows that $H^{*+V}_{(-)}(X;M)$ is represented by $F(S^V,HM)\simeq D(S^V)\wedge HM$ which is again ordinary. Here $DX=F(X,S)$ is the equivariant Spanier-Whitehead dual of $X$.

If you like, after fixing $M$ you can extend this into a functor from the subcategory of the homotopy category spanned by the representation spheres, their Spanier-Whitehead duals, and their smash products to the category of graded Mackey functors. One can try to take a skeleton of this category to obtain an '$RO(G)$-graded' functor, but this involves keeping track of the automorphisms and solving a coherence problem. My understanding is that this coherence problem can be solved, but that it involves choices (which never seem to be specified).

As Peter pointed out the representation spheres are only distinguished by the fact that they are invertible, well understood, and play a pivotal role in the duality theory for $G$-manifolds. The canonical choice from a homotopical perspective would be to think of the Picard group of invertible objects instead.