Let $(M,g)$ be a compact Riemannian manifold with strictly convex boundary.
Let $\gamma:S^1\to M$ be a periodic billiard trajectory (geodesic in the interior and reclects specularly at the boundary).

For some small $\epsilon>0$, let $g_s$, $s\in(-\epsilon,\epsilon)$, be a family of metrics on $M$ so that $g_0=g$.
Assume that all metrics and their dependence on the parameter $s$ is smooth, but I don't want to constrain the variation of the metric otherwise.

**Is there a (smooth) family of closed curves $\gamma_s:S^1\to M$ (with some $\epsilon>0$) so that each $\gamma_s$ is a periodic billiard trajectory with respect to the metric $g_s$ and $\gamma_0=\gamma$?**
(With a suitable notion of smoothness at reflection points.)
In some cases a perturbation of the metric can destroy the periodic trajectory (consider the 2-periodic trajectory on a square), but it seems to me that strict convexity of the boundary should add stability.

The most interesting case to me is when $M$ is three dimensional, in particular when $M$ is the unit ball in $\mathbb R^3$ and the metric $g_0$ is conformally Euclidean, but I would like to understand the problem in greater generality.
Section 7.1 [in this chapter](https://www-fourier.ujf-grenoble.fr/~lanneau/references/masur_tabachnikov_chap13.pdf) (for example) discusses this question for planar polygons, but that is quite far from my goal.

In principle one could glue two copies of $M$ together along the boundary and study the stability of periodic geodesics on the closed manifold.
The problem is that the metrics on the doubled manifold are not even $C^1$ (only Lipschitz) because $\partial M$ is strictly convex, so I fear that many tools are not applicable.
The normal derivative of the metric tensor is the second fundamental form, so the following are equivalent: (1) the metric is $C^1$ (2) the metric is $C^2$ (3) the second fundamental form vanishes.

**Edit:**
Carl's answer below tells that most periodic orbits are not stable if the unperturbed manifold $(M,g_0)$ is too symmetric.
This is a useful answer but only to a special case.
Are the orbits stable for generic metrics or under some additional assumption?