Induced map on path manifolds: is it a submersion? Consider the following claim: 

Let $p:M \to N$ be a (surjective) submersion of finite-dimensional smooth
  manifolds. Let $J$ denote one of $[0,1],\ [0,1),\ (0,1]$. Then $p_*:M^J \to
N^J$ is a submersion of Frechet manifolds, where $X^J$ denotes the usual manifold of smooth paths in $X$.

The intuition is that given a lift of a path $\gamma$ through $p$, one can find
a neighbourhood $U$ of the image of $\gamma$ and a neighbourhood $V$ of the image of
the lift such that for every path in $U$ one can smoothly choose a lift to $V$.
Here I suppose we need a submersion of Frechet manifolds to be a map that
admits local sections through every point in the domain, if that is
the 'correct'
notion of submersion in that setting (certainly not the 'surjective on tangent
spaces' version).
I think the proof would use the characterisation of submersions as maps
which look locally (on both the domain and codomain) like projections
$U \times \mathbb{R}^n \to U$, and the existence of good open covers with smooth
contractions.
I think I'm able to prove that there are continuous sections through every point
in the domain, thinking of everything as a topological space, and using the
compact-open topology on the mapping spaces. But I don't know off the top
of my head that the compact-open topology on the space of smooth paths
is the same as the topology inherited from the Frechet manifold structure.
(My guess is that it is.)
My question: is the claim true?

As Andrew Stacey points out in the comments, the mapping space is not a manifold for non-compact intervals. However, I think I really only need maps which have all derivatives uniformly bounded (but a different bound for each derivative!). Since the topology on the mapping space for compact intervals uses uniform convergence, I'm betting that this set has the structure of a Frechet manifold.
Question 2: am I right?
Question 1': if so, is the claim true for this (putative) map of Frechet manifolds?
 A: 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.
A: I think I can answer my own question:
First, the issue of the (half-)open intervals versus closed intervals. There is a Frechet space of sections of a vector bundle over a manifold $X$ (not necessarily compact!), where we require that the sections have bounded derivatives of all order. Then the usual family of seminorms (sums of sups over $X$ of norms of derivatives up to order $n$) is well defined. Then we can consider the set $M^{J,b}$ of paths (using $J$ as above) with bounded derivatives (which is all paths in the case of a compact interval), and give local charts which are of the form $(\mathbb{R}^m)^{J,b} \simeq \Gamma_b(J,\gamma^\ast TM)$, sections (with bounded derivative) of the trivialisable vector bundle $\gamma^\ast TM \to J$. The usual arguments should (I haven't check all the details, but there don't seem like any obstacles) give the result that $M^{J,b}$ is a Frechet manifold, which reduces to the usual manifold of paths in the case that $J$ is a closed interval.
So in what follows we will only consider paths and sections both with bounded derivatives, but will drop the 'with bounded derivatives' for brevity.
Let $f:M \to N$ be a surjective submersion. There exists an open cover $U=\coprod_I U_i$ of $M$ such that $U_i = \mathbb{R}^m \times V_i$ for an open cover $V = \coprod_I V_i$ of $N$ and the induced map $U \to V$ is projection on the second factor. Thus the induced map on tangent bundles $TU \to TV$ is split.
Let $\gamma:J \to N$ be a path and $\gamma':J\to M$ be a lift through $f$. Then the induced map $f_\ast$ restricts to a map on charts
$$
\Gamma_b(J,\gamma'^\ast TM) \to \Gamma_b(J,\gamma^\ast TN)
$$
The local splitting of the map of tangent bundles $TM \to TN$ gives rise to a local splitting of the map (over $J$!) of tangent bundles.
Now consider a pair of vector bundles $E \to J$, $F\to J$ (the following argument works over any manifold, and lets us generalise the result from $J$ to an arbitrary finite dim, paracompact manifold), a map of vector bundles $E\to F$, and an open (numerable) cover $c:U\to J$ such that $c^\ast E \to c^\ast F$ is a split map of vector bundles over $U$. Then the following chain of maps gives us a section of the map $\Gamma_b(J,E) \to \Gamma_b(J,F)$ of Frechet spaces:
$$
\Gamma_b(J,F) \stackrel{c^\ast}{\to} \Gamma_b(U,c^\ast F) \stackrel{\sigma}{\to} \Gamma_b(U,c^\ast E) \stackrel{\sum\phi\cdot}{\to} \Gamma_b(J,E)
$$
where $\sigma$ is a splitting of $c^\ast E \to c^\ast F$, $\phi$ is a partition of unity subordinate to $U$ and the last map takes a family of sections $s_i$ and sends it to $\sum_I \phi_i\cdot s_i$.
Thus there are charts of the path spaces such that the map in question, $f_\ast:M^{J,b} \to N^{J,b}$ looks like a split projection when restricted to those charts, hence is a submersion if we take the definition given in Hamilton's "The inverse function theorem of Nash and Moser".
In particular, for the case when $J$ is a compact interval we have a submersion of the usual path manifolds.
A: A different version of this result, inspired by Andrew's paper, is Lemma 2.4 in

*

*Habib Amiri, Alexander Schmeding A differentiable monoid of smooth maps on Lie groupoids, Journal of Lie Theory 29 (2019), No. 4, 1167–1192, arXiv:1706.04816.

It uses more classical methods of global analysis and differential geometry, rather than generalised smooth spaces and smoothly-local methods.
