Another perspective is that the object's effective mass is going exponential as v->c. I mean, that's why we say "rest mass", right?
For a moving object you could then m=(Er+Ek)/c^2, which creates the impression that the mass is variable (as the term Ek is zero when at rest and increasing with velocity), giving rise to the terms 'rest mass' and 'relativistic mass' respectively for the rest energy and total energy equations.
This interpretation is somewhat outdated but the terminology rest mass maintains its legacy. One could refer to it as the `(proper |invariant |intrinsic )?mass` instead.
The variable mass issue is then 'solved' by 'refactoring' the equation to use momentum where mass is coupled with velocity, over which the complexity of the lorentz factor is engulfed.
If we cancelled out that movement (relative to the Milky Way), does the mass change?
Is how much do the various relative movements affect our mass, and would it be possible to pull tem apart? (Solar System, Relative to galactic center, other galaxies, etc)
What I understand is that the contemporary conception is that the mass does not change.
I just presented the argument for a notion of rest mass and relative mass.