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.
On the surface, it does seem like there is a similarity. If a particle is measured over here and later over there, in quantum mechanics it doesn’t necessarily have a well-defined position in between those measurements. You might then want to liken it to a game engine where the particle is only rendered when the player is looking at it. But the difference is that to compute how the particle arrived over there when it was previously over here, in quantum mechanics, you have to actually take into account all possible paths it could have taken to reach that point.
This is something game engines do not do and actually makes quantum mechanics far more computationally expensive rather than less.