Let's try this: the cost of doing an MMX on the Moon would be stupendous, at least US$100 million I would guess.

There are a zillion competing ideas on how to spend US$100 million, on doing good science.

The manesiro idea* has no math, no quantitative predictions (just that such an experiment might yield a non-NULL result, whatever that is); nothing has been offered to show that the thousands (millions?) of experimental results that show consistency with SR have been misinterpreted; nothing to replace SR (in the sense of successfully predicting the same results for those experiments which tested SR); and so on.

Now, please remind me again why it's worth so much money to do such a test?

*As far as I know, it hasn't even been published anywhere yet, so no one - other than manesiro - even knows what it is!