what would the mass and density of Sirius A be if the mass of Sirius B is roughly the same as the Earth?
The masses of the two objects are completely determined by a careful measurement of their relative positions on the sky over decades, combined with an accurately known distance.
There is no degeneracy in this solution. For a given orbital period and separation of the two objects, the masses are entirely determined. If Sirius B were not a massive white dwarf then the orbit would be different.  Specifically, if you made the total system mass smaller, but with the same orbital period, then the size of the orbit would be smaller. If you tried to compensate by adding mass to Sirius A, then it would almost coincide with the centre of mass of the system and would show almost no orbital motion.
The density is then found by combining these masses with radii estimated from the measured luminosities and temperatures of the components and is therefore also fixed (to about $\pm 10$% because of the observational uncertainties in the mass and radius determinations).