Prove that the composition map is continuous with respect to the metric topology on $operatorname{Iso}(M)$

Let $M$ be a finite dimensional Riemannian manifold and $operatorname{Iso}(M)$ be its set of isometries. It can be shown that $operatorname{Iso}(M)$ is a finite dimensional manifold with a metric as defined below:

Consider $(n + 1)$ points on $M$ so close together that $n$ of them lie in an normal neighborhood of the other, and if the points are linearly independent (i.e. not in the same $(n-1)$-dimensional geodesic hypersurface). Then the distance $d(f, tilde f)$ between two isometries $f$ and $tilde f$ will be defined as the maximum of the distance $d_i[f(x), tilde f(x)]$ as $x$ ranges over the given set of $n+1$ points. This distance can be shown to satisfy the usual metric axioms. Here $d_i$ is of course the induced metric on $M$ (Riemannian distance fucntion)

Given $operatorname{Iso}(M)$ is now a metric space with metric $d$ as defined, we thus get a natural metric topology for $operatorname{Iso}(M)$. That is open sets are all subsets that can be realized as the unions of open balls of form $B(f_0, r) = {f in operatorname{Iso}(M): d(f_0,f)< r}$ where $f_0 in operatorname{Iso}(M)$ and $r>0$.

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I am trying to prove that $$mathscr M: operatorname{Iso}(M) times operatorname{Iso}(M) rightarrow operatorname{Iso}(M), , (f,g) mapsto f circ g$$ is continuous in the metric topology of $operatorname{Iso}(M)$.

Attempt: Munkres Topology Section 46 Page 287

Let $Y$ be locally compact Hausdorff, and $X$ and $Z$ general spaces. Also let $mathscr{C}(X,Y),,mathscr{C}(Y,Z),$ and $mathscr{C}(X,Z)$ denote the spaces of continuous functions from the respective spaces with the compact open topology. Then the composition map
$$mathscr M: mathscr{C}(X,Y) timesmathscr{C}(Y,Z)rightarrowmathscr{C}(X,Z)$$
is continuous.

The above is a proven statement, and can be assumed for now. In the statement, $X$ and $Z$ can be replaced with $M$ which has metric topology (and hence manifold topology) and thus is a general space. Further, $Y$ can also be replaced with $M$ as it is locally compact Hausdorff as a manifold. So we end up with $mathscr{C}(M,M)$ for all 3. Furthermore, as isometries are continuous, we get that $operatorname{Iso}(M) subset mathscr{C}(M,M)$. Thus we end up with the following:

$mathscr M: operatorname{Iso}(M) times operatorname{Iso}(M) rightarrow operatorname{Iso}(M)$ is continuous

Idea: The compact-open topology and metric topology are the same in case of $operatorname{Iso}(M)$ under these conditions because the topologies of every space involved here comes from same $d_i$ (Riemannian distance function as defined previously)

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Q) So I’m looking for a proof that the CO Topology and metric topology are the same for $operatorname{Iso}(M)$.

Alternatively (and preferably)

Q) Is there a direct way to show continuity of $mathscr M$ in the metric topology of $operatorname{Iso}(M)$ (i.e. showing inverse of an open set in metric topology of $operatorname{Iso}(M)$ is always open in $operatorname{Iso}(M)timesoperatorname{Iso}(M)$, or any of the equivalent definitions of metric continuity) ?

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