In some respects this is unsurprising, as you have to use relativistic engineering to get a synchrotron to work right in the first place: but when you do get a synchrotron running, you will observe that there is a characteristic "synchrotron radiation" and indeed one of its properties is that it's 'strongly collimated' -- it points in one direction like a laser, not in many directions like a light bulb. And, the direction that it points is tangent to the circle in the direction that the charged particle is travelling, just as the headlight effect says it must be.
So in fact this now goes way beyond theory and is now a part of practical engineering, as a part of available laboratory and medical devices. So-called synchrotron light sources are not just testing this prediction and finding it resoundingly true, but indeed we're way beyond that and using it to build highly-useful devices for crystallography and medical imaging and the like. X-ray tubes using Bremsstrahlung ("braking radiation", you fire an electron gun into a big spinning plate and the electrons emit X-rays in all directions as they come to rest relative to that plate) still exist and are much cheaper sources for lower-energy X-rays, but when you want these highly-tunable, coherent, collimated sources these more-expensive devices are the state of the art.