I actually don't think the formation of a neutron star or a black hole involves a catastrophe in the force balance, though it is popular to explain it that way. To me, a catastrophe in the force balance would require that there be no hydrostatic solution, say via a bifurcation, even when we treat the system as not undergoing any thermal changes, i.e., not losing heat to its surroundings and not undergoing internal changes in composition. Creation of a neutron star is often characterized as what you get when you have a white dwarf at zero temperature whose mass exceeds the Chandrasekhar mass-- there is indeed no hydrostatic solution for such an object, so we might characterize its collapse as a catastrophe. However, this is an artifact of the questionable assumptions, because there is no reason to assume you have an object that maintains zero temperature in a real core-collapse scenario. Instead, you have a mass, which can exceed the Chandrasekhar mass, and you have an internal energy density, which because of the history of the object can certainly yield a nonzero temperature. That object can have a hydrostatic solution if the energy density is high enough, and indeed it seems to me it normally will be, because objects like this are generally created by dropping in mass that either already has a high energy content, or falls in from a far enough distance to acquire a high energy content. So I think it would be pretty unexpected to ever arrive at a state that does not have a hydrostatic solution, such that it could be said to collapse as a true catastrophe.
If that is indeed true (and an alternative possibility is that the general relativity of the situation leads to unstable orbits such that hydrostatic equilibrium is indeed impossible), then the core collapse could involve an object that has a hydrostatic solution that is possible, for its given energy content, if that object could just be left alone long enough to find that solution. But the object is not left alone, it is undergoing change that comes either from loss of heat via neutrino losses (perhaps the URCA process), or from photodisintegration of its iron that is changing its internal composition. When the gas is highly relativistic, even if it does have a hydrostatic solution if none of these changes were occurring, the rate at which the hydrostatic structure would need to reconfigure to adapt to these changes is strongly leveraged by the fact that relativistic gas is very soft-- by which I mean simply that large reconfigurations are necessary given even small internal changes in composition or heat content. At some point, these reconfigurations are so leveraged that they take much longer than just the sound crossing time, and meanwhile the thermal timescale is getting shorter and shorter because it could in principle get as short as the light crossing time. When the thermal timescale gets shorter than the force-balance timescale, you get free fall, but not because there exists no instantaneous hydrostatic solution, it is because the hydrostatic solution takes too long to achieve. So the catastrophe is a thermal instability, not a catastrophe in the force balance. The force balance contributes to the problem only by how soft it is-- small thermal changes lead to large reconfigurations which drive further thermal changes, until the thermal physics runs away. Take away the thermal changes and the force balance might have no problem finding a continuously changing configuration as you add mass particle by particle, you would just drive a steady and continuous transition to a neutron star that could be made as slow as you like by adding particles as slowly as you like. The thermal runaway precludes that.
So what I'm saying is, I suspect that if you added mass gradually to a white dwarf, then before you even quite get to the Chandra mass you will start to see the thermal adjustment timescale becoming comparable to the force-balance establishment timescale, and when the two are about equal, you will lose control of the process-- it will take on a life of its own that no longer cares how fast you are adding particles, and will lead to a collapse, not because there is no force balance (as is true above the Chandra mass), but because there is a thermal instability that operates faster than the forces can equilibrate. That's not a formal catastrophe in the sense of a bifurcation in which the force balance is lost, but it is a catastrophe in the sense that it is a thermal runaway.
Of course, in real stars, we don't know how fast the mass is being added-- it might be added abruptly, like in a white-dwarf merger. But in that case, as I said above, there's still no reason to assume the temperature will be zero, so there could still be a force balance that is possible at the higher mass. You'll still have to wait for the heat to get out, so you will still be controlled by the thermal timescale, and the catastrophe will still happen when the thermal timescale gets short enough to rival the force-balance timescale. So I think that will be true whether you add mass slowly or abruptly-- the core collapse is a thermal instability that plays out in situations where there generally would be a hydrostatic solution, but it simply does not have time to set up given the rapidly changing thermal environment. The criterion for the catastrophe is equating the thermal adjustment rate to the relativistically softened force-balance rate, which happens before you get to the Chandra mass and does not matter how fast you add the particles once you get to that critical limit.