It shouldn't be that hard to figure out quantitatively. Instead of simple path reflection round-trips, you set up a triangle, say from Earth to moon to a satellite that is on orbit around Earth, Moon or Sun. Comparing the round-trip straight line of the moon reflector with varying alternate return leg vectors and timing should determine if there is anisotropy in space-time directions.
Same thing with the clock synchronizations. If you move the clocks away from center at differing speeds and/or to unequal distances, then you have variables to work with that aren't simple round-trip cancellations. If there is a variable that isn’t overwhelmed by the influence of Sol system gravity web, then it should show up at some point in the motion of Earth, Sol and the Milky Way through space-time.
I would think that someone by now would just build a clock with inertial sensors and program it to self-adjust based on ded reckoning accelerations (GPS satellites?) and mail it to the Royal for a prize.
Also, what Mike Scott said. (wow! there’s a lot of Mike Scotts! ha ha)