If you apply the more detailed equation I bet it says the mass doesn't change, so there is no change around the circle to measure.

I'm sure someone has thought of an experiment that simple. If you can't find anything close enough you can ask on one of the stack exchanges.

Well, the equation says that, unlike in newtonian physics, there's a gamma times the mass in the force equation. You can think of that as "the mass changing from the rest mass", or you can think of the mass as being constant and the gamma just being an additional factor, but either way, the gamma is still there.

Before I looked at stack exchange, I thought of another, much simpler experiment. Generate plane wave radio waves of a frequency such that the nominal wavelength would be meters or tens of meters. (By "nominal wavelength", I mean the wavelength l=c/f, the wavelength as if the speed of light were the same both directions.) Run those plane waves into a reflector a couple of nominal wavelengths away. Measure the RF energy at various points along the path to the reflector. Does it look like a standing wave of the expected wavelength, or not?

I actually saw that idea on the discussion I found on stack exchange. The only reply I saw was "well, the relationship between wavelength and frequency might not hold if the speed of light is asymmetrical", which seemed very weak to me. What, we have waves propagating with velocity v, but wavelength l =/= v/f? How can you do that without destroying the continuity of the wave? How much of physics is that going to destroy? And, how many "well, maybe..." items are you willing to stack up to make it impossible to detect your first "well, maybe"?

I didn't leave a question on stack exchange. The discussion was nine years old.