Induced map on path manifolds: is it a submersion?

This is answered affirmatively in Yet More Smooth Mapping Spaces and Their Smoothly Local Properties. Specifically:

Theorem 1.1

Let $M$ be a finite dimensional smooth manifold. Let $S$ be a Frölicher space with the property that there is a non-zero smooth function $C^\infty(S,\mathbb{R}) → \mathbb{R}$ with support in $C^\infty(S,(-1,1))$. Then $C^\infty(S,M)$ is a smooth manifold which is locally modelled on its kinematic tangent spaces.

Suppose, in addition, that $N$ is another finite dimensional smooth manifold and $f \colon M \to N$ a regular smooth map. Then $C^\infty(S,f) \colon C^\infty(S,M) \to C^\infty(S,N)$ is a regular smooth map.