The isomorphism, at least, exists. Take $M\to Y$ an injective resolution of $M$ as a $G$-module, and $\mathbb Z\leftarrow X$ a projective resolution of $\mathbb Z$ as a $G$-module. The double complex $\hom_G(X,\hom_{\mathbb Z}(Y,D))$ gives rise, as usual, to two spectral sequences. Computing first cohomology with respect to the differential induced by that of $Y$ and then by that of $X$, we get $H^\bullet(G,\hom(M,D))$. On the other hand, rewriting the complex as $\hom_{\mathbb Z}(Y\otimes_G X,D)$ using standard adjunctions, and then computing homology first with respect to the differential induced by that of $X$ and then by that of $Y$ gives the other side of your isomorphism. Convergence then does the trick.

That the isorphism is given by cap products should follow from general nonsense...

The isomorphism, at least, exists. Take $M\leftarrow Y$ and $\mathbb Z\leftarrow X$ projective resolutions of $M$ and of $\mathbb Z$ as $G$-modules. The double complex $\hom_G(X,\hom_{\mathbb Z}(Y,D))$ gives rise, as usual, to two spectral sequences. Computing first cohomology with respect to the differential induced by that of $Y$ and then by that of $X$, we get $H^\bullet(G,\hom(M,D))$. On the other hand, rewriting the complex as $\hom_{\mathbb Z}(Y\otimes_G X,D)$ using standard adjunctions, and then computing homology first with respect to the differential induced by that of $X$ and then by that of $Y$ gives the other side of your isomorphism. Convergence then does the trick.

That the isorphism is given by cap products should follow from general nonsense...

The isomorphism, at least, exists. Take $M\to Y$ an injective resolution of $M$ as a $G$-module, and $\mathbb Z\leftarrow X$ a projective resolution of $\mathbb Z$ as a $G$-module. The double complex $\hom_G(X,\hom_{\mathbb Z}(Y,D))$ gives rise, as usual, to two spectral sequences. Computing first cohomology with respect to the differential induced by that of $Y$ and then by that of $X$, we get $H^\bullet(G,\hom(M,D))$. On the other hand, rewriting the complex as $\hom_{\mathbb Z}(Y\otimes_G X,D)$ using standard adjunctions, and then computing homology first with respect to the differential induced by that of $X$ and then by that of $Y$ gices the other side of your isomorphism. Convergence then does the trick.

That the isorphism is given by cap products should follow from general nonsense...

The isomorphism, at least, exists. Take $M\to Y$ an injective resolution of $M$ as a $G$-module, and $\mathbb Z\leftarrow X$ a projective resolution of $\mathbb Z$ as a $G$-module. The double complex $\hom_G(X,\hom_{\mathbb Z}(Y,D))$ gives rise, as usual, to two spectral sequences. Computing first cohomology with respect to the differential induced by that of $Y$ and then by that of $X$, we get $H^\bullet(G,\hom(M,D))$. On the other hand, rewriting the complex as $\hom_{\mathbb Z}(Y\otimes_G X,D)$ using standard adjunctions, and then computing homology first with respect to the differential induced by that of $X$ and then by that of $Y$ gives the other side of your isomorphism. Convergence then does the trick.

That the isorphism is given by cap products should follow from general nonsense...