Any time you do something to the particles on Earth, the ones on the Moon are affected also
The no-communication theorem already proves that manipulating one particle in an entangled pair has no impact at al on another. The proof uses the reduced density matrices of the particles which capture both their probabilities of showing up in a particular state as well as their coherence terms which capture their ability to exhibit interference effects. No change you can make to one particle in an entangled pair can possibly lead to an alteration of the reduced density matrix of the other particle.
The no-communication theorem already proves that manipulating one particle in an entangled pair has no impact at al on another. The proof uses the reduced density matrices of the particles which capture both their probabilities of showing up in a particular state as well as their coherence terms which capture their ability to exhibit interference effects. No change you can make to one particle in an entangled pair can possibly lead to an alteration of the reduced density matrix of the other particle.