Is it possible that quasars are not powered by black holes but some other kind of compact, massive object?
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.