OK, this question is still bothering me, and I still don't know the answer.  Truth to tell, I suspect it is false.  

I write to point out that your two functors are adjoint.  Moer precisely, suppose we have a map of R-comodules from

A tensor_E M --> N

where M is an E-module.  Then we get an induced E-module map on the primitives 

P(A tensor_E M) --> PN

There is an obvious map M --> P(A tensor_E M)

that takes m to 1 tensor m.  Thus we get an E-module map M --> PN.

Conversely, if we have an E-module map M --> PN, then we get an R-comodule map 

A tensor_E M --> A tensor_E PN

then the multiplication map A tensor_E PN --> N is an R-comodule map, so we get 

an R-comodule map A tensor_E M --> N, and this makes the functors adjoint.