I recently came across Terence Tao's article https://terrytao.wordpress.com/2007/03/28/open-question-scarring-for-the-bunimovich-stadium/ on the scarring conjecture on Bunimovich stadium. Tao states that the Bunimovich stadium is not uniquely ergodic in the classical sense due to the bouncing ball trajectories. Unfortunately I have not seen many, if any, explicit examples of classically invariant measures which kill the uniqueness of the ergodicity on this stadium nor have I that much experience with ergodic theory that I could verify my answer: What are at least two such classically invariant measures which show that the billiard on the Bunimovich stadium is not uniquely ergodic? Is the following a correct construction for a family of such invariant measures?
Denote by $\Omega$ the Bunimovich stadium in question and suppose that we have chosen a fixed bouncing ball trajectory, call it $BB_1$, and that with discretized time steps the billiard flow along $BB_1$ contains some $N$ points $\{p_1,\dots,p_N\}$. Do we then just choose some constants $0 < c < 1$ and take $\mu_c = \frac{c}{N}\left(\delta_{p_1}+\cdots + \delta_{p_N}\right) + (1 - c)\tilde{\mu}$ where $\tilde{\mu}$ is the uniform probability measure on $\Omega\setminus\{p_1,\dots,p_N\}$ and $\delta_{p}$ is the Dirac delta at $p$?
It's simpler than what you are suggesting.
Using your notation with discrete trajectories, simply take the vertical bouncing trajectory described in that link you described, and take $\mu = \frac{1}{2}(\delta_{p_1} + ... + \delta_{p_2})$; here I am assuming that the two horizontal sides of the Bunimovich stadium are $1$ unit apart and the speed is $1$ so it takes $2$ units of time to bounce orthogonally between the top and the bottom of the stadium.
Alternatively, in the language used by Tao in that post, one of those vertical orthogonally bouncing trajectories goes back-and-forth-and-back-and-forth... along a single vertical segment, and this is certainly not uniformly distributed in either the billiard table itself nor in the phase space (the unit tangent bundle of the billiard table).