Despite some supporting references to physics, this question is purely mathematical on the calculation of volume in a curved metric space.
Consider a Schwarzschild space (spacetime) defined by a hollow thin spherical shell of the mass $M$ and the radius $r>R$, where $R=2M$ is the Schwarzschild radius in natural units given by setting physical constants to unity.
The exterior metric is Schwarzschild, the interior metric is time dilated Minkowski. Both are defined in details here (scroll down to “EDIT”): Can separate manifold regions have the same coordinates?
The volume inside the shell in tbe Schwarzschild coordinates is known for two extreme cases. (1) For $r\gg R$, the volume is Euclidean. (2) For $r=R$, the volume is zero according to The Volume Inside a Black Hole (chapter 3, pp. 5-6):
There is zero volume inside the black hole in any Schwarzschild time slice of a Schwarzschild black hole spacetime.
What is the volume inside the shell (as observed from outside) in other cases? I am interested in any scenarios, exact or approximate expressions or asymptotic cases. Thanks for your expert insight!