The OP seems to think the cosmic microwave background radiation (CMBR) was generated by the Big Bang singularity. This is incorrect. The CMBR was emitted by ordinary matter (plasma) that filled the universe (normal space as we know it) when the universe was young. Eventually the universe expanded and cooled to a point where the opaque soup of plasma turned into a clear space filled with hydrogen atoms. The radiation from the plasma has been flying around ever since, getting diluted by the expansion of space.
You had that backwards. The temperature of empty space is described as being 3 K because that’s the black body temperature the heavily red-shifted CMBR corresponds to. (And it’s an apt description, since that’s then the equilibrium temperature of an object in empty space.)