I was reading a question involving an ultracentrifuge to test General Relativity. Instead of using an atomic clock the asker posited using radioactive decay as the metric to evaluate time dilation effects. The answers did a good job of explaining that the acceleration due to rotation was more in the realm of Special Relativity so I wondered if the following experiment would be a valid / applicable test of General Relativity.
Take a quantity of a radioactive isotope, split it in half. One half stays on Earth, the other half is sent to the Moon, along with the appropriate detector(s). Since the gravity field experienced by the Moon is both different and smaller in total than the field on Earth, would the sample on the Moon decay at a slower rate and could that difference in rate be attributed to General Relativity? I'm assuming that the gravity field generated by the Moon's mass is less than the lessened field due to the increased distance from Earth.