I have a suggestion for you. Try it when A=k[G]$A=k[G]$ for a finite group G$G$ and E=k[H]$E=k[H]$ for a subgroup H$H$. Then R$R$ should be k[G/H]$k[G/H]$, which of course will only be a coalgebra and not a Hopf algebra if H$H$ is not normal. This example leads me to doubt your claim that R$R$ is a coalgebra in the category of A$A$-algebras, since I don't think R$R$ is an A$A$-algebra unless E is normal.
Anyway, your desired result should be something about induction and restriction in this case. Indeed, an E-module N is just a representation of H$H$. A tensor over E$E$ with N$N$ is just the induced G-representation.
