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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?

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In each reference frame the body has an energy $E$ given by its rest mass $m$ plus the relativistic three-momentum $p$. In natural units ($c$ = 1), it is $E = \sqrt{p^2 + m^2}$.
The energy is different in each reference frame because the momentum is different. In particular in the rest frame the momentum is zero.
If the body absorbs f.i. the energy and momentum of a photon, it will change both its rest mass and momentum, but the new energy and momentum will be different if measured in different reference frames. The new rest mass will be the same in each reference frame by definition.
Note: be careful not to confuse the mass with the relativistic mass, which is an obsolete concept just causing misunderstanding.

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