If I accept true mass as being constant then defining increases to such a given mass is impossible (see Newton) because I can sit and watch it forever at rest or in motion: a given mass is a given mass, and a given quantity does not change, regardless of speed or energy applied. The effect(s) then, either perception or theoretical, that creates the illusion that mass changes, is better described by a more accurate theory than Einstein put forth or modifying Einstein to a more accurate level.
On another note, the arbitrary Einsteinian speed of light limit for velocity is similar to the ceiling of the 4 minute mile. There is no proof that such a limit exists. Until we get past this arbitrary barrier and explain these phenomena in a more useful way we will never achieve deep space travel.
To make such travel possible we must get beyond mass particles into non-mass particles that pass easily through the light barrier. They might or might not exist at this point. In the future these particles will be the building blocks and the carriers of information necessary to transfer life to habitable planets throughout the universe. Granted, your body will not beam up but your current conscionsness and DNA sequence might: A quick resurrection and you are you three galaxies away.
Granted that I am not a physicist and I am an iconoclast when it comes to accepting dogma. To me, devolving honest perception into theoretical soup does not make you any more informed on the true nature of mass vis a vis energy and velocity. You simply have learned what someone else learned and thought it to be true, i.e. the appeal to authority that Einstein believed and said it therefore it is true.