To continue on the size of the theoretical radiation ball inside a black hole: The radiation will generate a pressure proportional to density and therefore should have the same density profile as a conventional gas star, varying about as 1/(r^2). The gravitation potentional energy of this density profile is about (2GM^2)/R.
If the results of the viral theorem can be used here (and maybe the viral therom can’t be used here because the supporting energy is always constant here regardless of size), the supporting energy would equal half the gravitational potential energy, or (2GM^2)/2R = (GM^2)/R. If the supporting energy is (1/3)Mc^2, R would equal (3GM)/(c^2), which doesn’t work because R then would be larger than the Schwarzchild radius. I think the most likely value for supporting energy would be (2/3)Mc^2 since the measured value of reflected radiation pressure is (2/3)Mc^2 from laboratory measurements. If the supporting energy is (2/3)Mc^2 and half the gravitational energy is (GM^2)/R, then the radius R of the radiation ball would equal (3GM)/(2c^2).
Note that (3GM)/(2c^2) is 75% of the Schwarzchild radius, which means a large observable burst of radiation should occur when 2 typical 8 solar mass black holes merge. I hope this is confirmed by observation in the future.