It's hard to give a definitive answer to this question. For the spherical-cow, Physics 101 answer, the answer is yes, the dynamics of orbital mechanics are time-reversible. However, as I'll get into later in my answer, for pretty much all practical scenarios that move beyond any simplified picture, the answer is no.
I think AtmosphericPrisonEscape really hit the nail on the head with their comment, so I can't take all the credit for this part of my answer. When one sets up the dynamical equations for orbital mechanics, and tries to solve them (be it with a Hamiltonian, Lagrangian, or something else), fundamentally they've built in the force governing the motion as the (Newtonian) force of gravity. I'd suggest taking a look at this question and answer for an example of setting up the equations of motion. In anycase, because we know the force of gravity is conservative (since it only depends on position and not on anything else, such as velocity), then the the orbital mechanics equations that are governed by the force of gravity are time-reversible. A conservative force allows one to perform an operation and then undo that operation and return to the same state. This statement is independent of how many bodies we have or what types of orbits they may have.
Now let's move beyond the spherical cow. Reality is not accurately modeled by the 101 orbital mechanics equations. Reality is more complex. There's all sorts of other processes at work such as gravitational radiation, yarkovsky effects, Poytnint-Roberston effects, drag within an accretion disk/solar wind/gas/dust/etc. (including drag from magnetic fields!), and many, many other effects that can cause orbits to deviate from the standard orbit one would calculate with the naive orbital dynamics as described above. So in that consideration, Kepler's laws, while generally true (by that I mean, it's often the "leading order term" in the list of all things effecting motions of a body), they are not exclusively true and don't 100% account for any given object's orbits.
However, it gets even worse! It may be that even in the spherical-cow, physics 101 case, the orbital motion equations may not be time symmetric. One often finds that in order to accurately predict orbital motions, they are relegated to using simulations (since it is known that anything more than a 2-body problem has no closed-form, analytic solution). Simulations come with their own issues. Computers do not have perfect precision or infinite computing power, which means concessions must be made. We must, necessarily compute with limited precision (that is, we may only be able to calculate numbers out to 15 digits) or with limited computing power (that is, we may only be able to compute the equations every 0.0001 seconds, but not to an accuracy better than that in time). What's more, different numerical techniques may have varying numerical stability. This leads to the concept of numerical dissipation or diffusion wherein, over time, small errors introduced by imprecision of the simulation results in a building inaccuracy of the simulation. This can cause your physics simulations to "bleed away energy" (hence the use of "dissipation" in the term above). This steady, stochastic loss of energy represents a time asymmetry meaning the simulation of orbital mechanics is always asymmetric in time.
I bring all of that up, as an interlude to the concept of Planck time and Planck length. Planck time and lengths are often seen as the smallest fundamental units of the universe. By that I mean, it doesn't make physical sense to talk about events happening on time scales shorter than a Planck time or on lengths smaller than a Planck length. This brings me to Boekhalt et al. (2020) who set out simulate three orbiting black holes with extreme precision (that is, they invested considerable computing resources to minimize the numerical dissipation as much as possible). The punch line of their research was that they managed to reduce their imprecision to below the Planck length and still found that a selection of their orbits were time asymmetric. If it is true that the Planck length is the smallest length at which physics makes sense, then they've shown that 100% time symmetry of orbital equations would require a precision greater than the universe would seem to allow.
So all of that is summarized by saying that yes, orbital dynamics are time symmetric if you don't consider all the things that can make them time asymmetric.