If you take $R$ to be, say, a finite field, so that all epimorphisms and monomorphisms of $R$-modules are pure and split, and + is just vector space duality, and for simplicity restrict to functors that take values in finite-dimensional vector spaces, then your question reduces to asking whether a monomorphism (or epimorphism) of such functors must be split.

This is certainly not true in general. For example, let $A$ be a finite-dimensional $A$-algebra, and $M\to N$ a non-split epimorphism of $A$-modules. Then $\operatorname{Hom}_A(N,-)\to\operatorname{Hom}_A(M,-)$ is a non-split monomorphism of functors from finitely generated $A$-modules to finite-dimensional $R$-modules.

If you take $R$ to be, say, a finite field, so that all epimorphisms and monomorphisms of $R$-modules are pure and split, and + is just vector space duality, and for simplicity restrict to functors that take values in finite-dimensional vector spaces, then your question reduces to asking whether a monomorphism (or epimorphism) of such functors must be split.

This is certainly not true in general. For example, let $A$ be a finite-dimensional $R$-algebra, and $M\to N$ a non-split epimorphism of $A$-modules. Then $\operatorname{Hom}_A(N,-)\to\operatorname{Hom}_A(M,-)$ is a non-split monomorphism of functors from finitely generated $A$-modules to finite-dimensional $R$-modules.