I'll leave my question here in case someone feels like answering.
We define an elementary particle to be a unitary irrep of the Poincare group, because this is the symmetry group corresponding to SR. The unitary irreps are labelled by mass and spin (m,s). But if a representation (aka a particle) has mass m, then haven't we already stepped outside the boundaries of SR (since the presence of mass implies we are not in flat space anymore)?