There is no such choice, if by "canonical" you mean natural in the category-theoretic sense. I am going to rename $V'$ to $W$ because I find different letters easier to parse.

You want to find a natural map $\text{Hom}(V \oplus V, W) \to \text{Hom}(V, W)$ with certain properties. Fixing $V$ and using only naturality in $W$, it follows by the Yoneda lemma that this is equivalent to finding a map $V \to V \oplus V$ and then applying $\text{Hom}(-, W)$. This is equivalent to specifying two maps $f, g : V \to V$ and considering the map $v \mapsto f(v) \oplus g(v)$. 

The only maps $V \to V$ which are natural in $V$ are given by scalar multiplication; this is a corollary of the fact that the center of $\text{GL}(V)$ is given by scalars, but this only uses naturality with respect to automorphisms and the proof is easier and more general if you make more thorough use of naturality (see <a href="http://qchu.wordpress.com/2012/02/06/centers-2-categories-and-the-eckmann-hilton-argument/">this blog post</a>). 

Anyway, the upshot is that the only natural maps from pairs of linear transformations $V \to W$ to linear transformations $V \to W$ are given by taking linear combinations; in other words, all you can do is take

$$(R, S) \mapsto a R + b S$$

for some fixed scalars $a, b$. It's not hard to show that no such choice does what you want.