While your idea of varying inertial reference frames is correct for invariant mass at a constant velocity (or experiencing no acceleration or distortion of spacetime due to a gravitational field), it is not correct when those are included.
In fact, it is easy to see that the relativistic mass must stay constant here for two reasons: 1) Relativistic mass can change between reference frames, but we are always doing the experiment in the same reference frame, and 2) Look at the equations. http://en.wikipedia.org/wiki/Lorentz_transformation#Lorentz_...
There are no gravitational terms in those equations, only spacetime derivatives (and various constants).
Edit: It's very late and I bet I'.m being unclear -- the velocity magnitude is probably changing slightly because of the outside fields. However, I doubt that relativistc mass change would be detectable by even their instruments. It's probably just variance in path.