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 this blog post).
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 (clearly both $a$ and $b$ must be nonzero and then you can take $S = - \frac{a}{b} R$).