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 (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?