Suppose you have a mass that is moving at relativistic speed wrt (with respect to) a powerful laser pointed at the mass. As the laser shines on the mass, assume it absorbs all the energy (and momentum), and therefore grows in mass (and it's momentum increases). The light undergoes a Doppler shift in the frame of the mass (but not in the frame of the laser), so in the frame of the mass it absorbs a different amount of energy than in the frame of the laser. Therefore, in each frame it has a different mass.
This does not seem possible, so does anyone know where my conceptual mistake is?