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.  
                             
                         Mark