Yes, two clocks, one stationary on the Moon and one stationary on the Earth. How many times I need to repeat this? 😄
Did you make (or read) proper GR calculations, or you are just guessing?
What reference frame was used and why?
With one clock on the Moon surface and the other on a Moon artificial satellite, yes, things would be exactly as expected. No reason to be different.
The test with one clock on the Moon surface and the other on the Earth surface was never done, nor any similar, so you don't really know the outcome. It is a dangerous thing, both in physics and in life, to be sure that you know something, when in fact you never really checked it.
Gravity is not different, that's not the issue, we simply have not enough/accurate information about "kinematic time dilation" caused by the movement of a planet/star/moon. If you do have such information, please share it here.
Loads of money? Tiny, compared to LHC, JWST, or gravitational-wave detectors.
So you don't expect that anything special would happen. I hope you are aware that a theory must pass any kind of test related to it. This kind of test (able to check with accuracy the "kinematic time dilation" caused by the movement of a planet/star/moon in relation with another planet/star/moon) is highly related to GR and never performed. If the test is not passed, GR would have to be adjusted or abandoned. You are denying that?