I don't understand this reasoning, the singularity at r=0 in the Scwharzschild solution itself is non-removable. It is a "real" singularity. So even if we made m constant, as is the case in the Schwarzschild metric, the singularity at r=0 is still there, we haven't removed it.
I think that (the r=0 singularity) is not for the energy momentum tensor... If m is constant the Einstein tensor vanishes, and so does the EM tensor.