If I am correct, the particle accelerated to near light speed does not actually gain mass, it has an apparent mass gain when observed from outside because it's time has slowed down. I have decided that the choice between more mass/less time is indistinguishable outside of the system under study. Since both sets of equations reduce to the same expression, one cannot look at this case to determine what is really going on.
Okay, let’s take a simpler situation lacking relativistic speeds. Mass can be observed to be lost in its own reference frame, say, when deuterium-tritium fusion takes place. There’s a certain amount of mass beforehand, and measurably less after the fusion takes place. Thus, mass is not conserved. I expect you’ll have a tough time reconciling your theory with this.