Yeah, the “real” version of Newton’s 2nd law, as in the version that’s true and generally applicable, is “Force equal rate of change of momentum”, or symbolically F=dp/dt. That is the definition of “force”. You’ll never see a proof of Newton’s 2nd law, because it is not a theorem but actually a statement of definition.
So, once you know that force is defined as F=dp/dt, then calculating force just means taking the derivative of momentum. And as it turns out, in special relativity the momentum of an object is p=γmv, where γ is the Lorentz factor γ=dt/dτ=1/√(1-v2/c2).