AFAIK, objects in the universe thought to be black holes show evidence of radiatively inefficient accretion flow. But is this the case with Quasars? If not, is there a possibility that the core of a quasar is an object similar to a neutron star.
The solution to your question is surprisingly simple, I think:
A quasar that puts out energy around Eddington luminosity or higher, must accrete at a certain rate, corresponding to the energy output.
When consulting a lecture and a random paper from the archive on this topic, it is evident that those accretion rates correspond to 1-10 solar masses per year or higher. A neutron star accreting 1-10 solar masses in one year will become a very massive black hole quickly enough.