So, assuming we have a massive ball of water that keeps growing, but somehow manages to remain at a fixed density, the moment the Schwarzschild radius overtakes the physical radius, will the gravitational properties of the ball of water undergo a sudden, dramatic change?

Here explains in a bit detail.In here It says its possible.But I dont think it would be sudden (Not sure though), cause we are adding matter in some amount so probably as we add the matter things will change in a someway, like gravity affect on the center will increase an amount so it will keep increasing with addition of matter and finally it will collapse I guess

There is nothing magic about black holes; they are just massive objects. They obey the same laws of gravity other every-day objects do. So from a distance, when you add that last bit of water, nothing noticeable will change about the gravitational field.

I don't know what you mean by "the region", but notice I said "nothing noticeable will change about the gravitational field". What a distant observer sees with his eyes will change.

Are you above or below the Schwarzchild radius? Are you a point particle or a person who is more than 20cm tall? Trying to descipher non-physical assumptions is hard...

While I don't know much about GR and black holes, I would say to an outside observer all Schwarzschild BHs of a certain size (mass) look identical no matter how they formed. In fact the mass is the only independent parameter needed to fully describe them.

As for a ball of water I don't think it can possibly maintain any finite density as the event horizon surpasses its diameter, it will have to collapse into a singularity. Nothing inside an event horizon can be static, the inside can only be a vacuum with a central singularity.

If you have a perfect fluid in hydrostatic equilibrium at some radius 20 cm greater than the Schwarzschild radius then there is no event horizon. You can go down to the middle and come back up again.

Note, the fluid in question cannot be water due to Buchdahl's theorem, it would have to be some unobtanium. The tidal forces would be enormous so you would have to be made of a different unobtanium.

As the radius becomes less than 9/8 of the Schwarzschild radius the pressure st the center becomes infinite and even your unobtanium fluid collapses. As you get near that limit the system becomes unstable and even your treading water will cause the collapse.