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.