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