Here is how an inertial frame of reference can be established for the case of a double star, using only the motion of that double star system.
The two stars making up that double star system are orbiting each other. Both stars are moving, but there is a point that is not accelerated: the common center of mass of the two stars. The two stars are orbiting the common center of mass.
The orbital motion of those stars doesn't have to be circular orbit. The orbital motion can be in the shape of an ellipse. The general case is called 'Kepler orbit'.
To prepare for the following I recapitulate some properties of the mathematical object 'ellipse': an ellipse has a major axis and a minor axis, and those to axes are perpendicular to each other
The shape of a Kepler orbit is an ellipse. A Kepler orbit has the following property: the orientation of the axes remains the same.
Conversely, you can use that property to find the inertial frame of reference, using only information from within that star system. The inertial frame of reference is that coordinate syste where the axes of the orbits remain in the same orientation.
Our own solar system is a single star system, of course, but we can still apply the same type of reasoning.
If you use a non-rotating coordinate system for representing the motions of the planets, then all planets move according to a single law of gravity: Newton's law of gravity.
What would you get if you would use a coordinate system that, unknown to you, has a small rotation rate? Then the orbits of the planets would not line up. You would need to fudge the orbit of each planet, and for each planet the fudging would be slightly different from the other planets.
The reason I'm describing this:
In order to measure which coordinate system is the non-rotating coordinate system for the solar system it is sufficient to use orbit data from the solar system only; that's enough.
Historically all planetary motion has been measured with respect to the background of stars. But if the solar system would be inside some interstellar dust cloud that would hide all distant stars, even then astronomers would eventually find the universal law of gravity, allowing them to find from measurement the inertial frame of reference of the solar system as a whole.