Could a quasi-star potentially develop a stable outer layer or crust during its early stage of formation if it were to be bombarded with substantial amounts of matter, coexisting with its core and the central black hole?
Despite the inherent instability of quasi-stars, is it theoretically possible for such a multi-layered structure to exist?
If by crust you mean a solid surface, then no, it would not be possible. Wikipedia states that the surface temperature of a Quasi-Star would be around 10,000 K. However, Hafnium Carbonitride, a compound made of Hafnium, Oxygen, and Nitrogen, has a melting point of 4,270 K. This is the highest known melting point known so far. So, if by crust you mean solid surface, then no material known to humankind would be able to not melt under those high temperatures. As well as this, if the crust blocks outgoing radiation, it would heat up to immense temperatures, well over 10,000 K. If the crust is opaque, then something similar to the Kappa Mechanism could occur. The crust will be pushed out, making it impossible for a stable solid surface to exist on the surface of a Quasi-Star.
