I think you're asking for experiments in which there are two different physical reference frames, i.e., two arrays of Einstein-synchronized clocks in relative motion. The muon experiment doesn't count because there are no sets of comoving muons with Einstein-synchronized decay times.

You're right, it's hard to think of such experiments. Normally there's one convenient reference frame (the lab frame) and constructing another is prohibitively difficult.

It doesn't matter because relativity isn't about reference frames, even though they were a big part of Einstein's original paper and have dominated introductions to it ever since. It's really about spacetime geometry. In Euclidean geometry, you can do some constructions without any coordinates at all. In many other cases it's easier to set up a Cartesian coordinate system. But it's rarely helpful to use two different Cartesian coordinate systems when solving the same problem. You don't need to, since one coordinate system gives you coordinates for everything, and that's enough to do the algebra.

The crucial thing in special relativity is the distance formula $\sqrt{Δt^2-Δx^2-Δy^2-Δz^2}$, which plays the same role as the Pythagorean formula. Lorentz transformations are important only because they preserve that formula. If you use rapidity (spacetime angle) instead of velocity (spacetime slope) in the Lorentz transformation, it looks like a transformation between Cartesian coordinate systems with a common origin. If you rewrite the Cartesian formula in terms of slope, it looks like a Lorentz transformation.

If we did an experiment with a second set of Einstein-synchronized clocks, it could still be analyzed from the perspective of a single reference frame (probably the lab frame). The results would be consistent with that analysis. And the results wouldn't prove anything, because you could still argue that the lab frame is the only true frame and special relativity only appears to be true because of distortions of the moving clocks, etc., just as everyone believed before 1905. The universe doesn't know what a clock is or what Einstein synchronization is, so it would just follow the same rules as always.

I think you're asking for experiments in which there are two different physical reference frames, i.e., two arrays of Einstein-synchronized clocks in relative motion. The muon experiment doesn't count because there are no sets of comoving muons with Einstein-synchronized decay times.

You're right, it's hard to think of such experiments. Normally there's one convenient reference frame (the lab frame) and constructing another is prohibitively difficult.

It doesn't matter because relativity isn't about reference frames, even though they were a big part of Einstein's original paper and they've dominated introductions to it ever since. It's really about spacetime geometry. In Euclidean geometry, you can do some constructions without any coordinates at all. In many other cases it's easier to set up a Cartesian coordinate system. But it's rarely helpful to use two different Cartesian coordinate systems when solving the same problem. You don't need to, since one coordinate system gives you coordinates for everything, and that's enough to do the algebra.

The crucial thing in special relativity is the distance formula $\sqrt{Δt^2-Δx^2-Δy^2-Δz^2}$, which plays the same role as the Pythagorean formula. Lorentz transformations are important only because they preserve that formula. If you use rapidity (spacetime angle) instead of velocity (spacetime slope) in the Lorentz transformation, it looks like a transformation between Cartesian coordinate systems with a common origin. If you rewrite the Cartesian formula in terms of slope, it looks like a Lorentz transformation.

If we did an experiment with a second set of Einstein-synchronized clocks, it could still be analyzed from the perspective of a single reference frame (probably the lab frame). The results would be consistent with that analysis. And the results wouldn't prove anything, because you could still argue that the lab frame is the only true frame and special relativity only appears to be true because of distortions of the moving clocks, etc., just as everyone believed before 1905. The universe doesn't know what a clock is or what Einstein synchronization is, so it would just follow the same rules as always.