Using the Dirac Equation, which is relativistically covariant let's you write down the Hamiltonian directly in any reference frame you like. Case closed. The interesting point of the derivations in the non-relativistic realm is that you get, even for an elementary particle as the electron, where the Dirac Equation is applicable, you get gyro factor that is wrong by a factor of 2 when using the Galilei transformation at this point of the derivation. This is known as Thomas precession and marks the discovery of the Wigner rotation in the composition of two Lorentz boosts that are not in the same boost direction, which is completely missing in the Lorentz transformation.
Of course, the true gyro factor of the electron (and the other leptons) is not exactly 2 but (g-2) is one of the most accurately measured quantities which are (for the electron) in astonishing agreement with QED (at the four-loop order of perturbation theory!), while for the muon there may be a discrepancy with the standard model since the most accurate measurement of (g-2) performed at BNL shows a deviation by somewhat more than 3 standard deviations. This is not a discovery but only an evidence for physics beyond the standard model, and for this reason the entire experiment has been moved to Fermilab now and will be driven to even higher precision very soon. The main theoretical problem is the contribution from the strong interaction to the radiative corrections, which are somewhat larger for the muon than for the electron. These corrections have thus to be measured with high accuracy too. Such experiments at very low energies (in contrast to the largest energies in the experiments at LHC) are done in Mainz, and also their efforts towards even higher precision is under way. It's a very exciting issue in elementary-particle physics!