So what is the inertial frame of reference in which orbital motion happens bound to?
TL;DR: Whatever you want. Conceptually, all frames of reference are equally valid. However, some frames are computationally better than others, depending on context. While you could do it, it would be ludicrous to describe the formation of a hurricane or the orbit of a satellite in low Earth orbit from the perspective of a Triton-centered, Triton-fixed frame.
But you could do it!
In fact, modeling the orbit of an object in low lunar orbit from the perspective of a Neptune-centered inertial frame is one of my favored tests of the orbital mechanics package I developed for the Johnson Space Center. The result is pure garbage after a few dozen orbits, but it does work initially. The object initially orbits the Moon, but numerical issues quickly arise.
What is this thing that you call an "inertial frame of reference"? As a supervisor said to me almost 40 years ago, name one. The so-called Earth-centered inertial frame obviously isn't inertial; the Earth is accelerating gravitationally toward the Sun, the Moon, the other planets, nearby and remote stars, other galaxies, etc. In addition, the axes of an ECI frame are almost certainly rotating with respect to those of a Newtonian inertial frame of reference.
There's one catch, good luck finding a Newtonian inertial frame of reference. Or as my supervisor said almost 40 years ago, name one. To make matters worse, this doesn't even take general relativity into account. Ultimately, the concept of a Newtonian inertial frame of reference is a fiction. That said, it is a very, very useful fiction because our solar system is very close to Newtonian in behavior. Even the motion of Mercury can be approximated extremely accurately as being due to Newtonian gravity plus some very small post-Newtonian accelerations.
There are two challenges with regard to defining a Newtonian frame of reference, the placement of the origin and the placement of the axes. It's important to keep in mind that all frames of reference are equally valid. Using a quasi-inertial solar system barycenter frame to describe the motion of a satellite in low Earth orbit doesn't make any sense. An Earth-centered inertial perspective is a much more sensible perspective.
As previously mentioned, an Earth-centered frame is an accelerating frame. This is easily addressed: Add fictitious accelerations due to the Earth's gravitational acceleration toward the Sun, the Moon, and perhaps the other planets. In the space exploration community, the term describing perturbations due to choosing a frame based on the center of some massive body is "third body effects" (or third body accelerations, or third body perturbations).
A rotating frame makes for a much messier situation. Up until the mid 20th century, the preference was to use a very slowly rotating frame based on the location of the vernal equinox. This resulted in apparent apsidal precessions of the orbits of the planets about the Sun. As is the case with third body effects, this is not necessarily problematic. The techniques that addressed this apparent precession were sufficient for the 19th century discovery that Mercury suffered a precession that could not be explained by Newtonian mechanics.
Three key things changed in this regard during the latter half of the 20th century. One was that humanity started putting things into space. Another was drastic improvements to astronomical observations. Both motivated the improvement of the concept of frames of reference.
The third key item was the discovery of quasars. Quasars are so remote that their proper motions are are extremely small and are unrelated to anything close by. (A galaxy even remotely connected to the Milky Way qualifies as "close by" compared to quasars. The current gold standard with regard to the orientation of a frame of reference is the International Celestial Reference Frame (ICRF). This is based on almost 300 quasars, with over 3000 other quasars used as a sanity check.