I would very much like to know what your source was for the above remarks about the moment of inertia of a solid sphere.
There is some information online which would allow you to calculate the moment of inertia of a relativistic hoop (and perhaps a disk) using some assumptions about the material, i.e. a so-called hyperelastic model, at:
http://www.gregegan.net/SCIENCE/Rings/Rings.html
This is based in part upon an online discussion here at
https://www.physicsforums.com/showthread.php?t=168121
However, I am not aware of anyone textbook or article which goes through a similar derivation, nor am I aware of anyone analyzing the case of a sphere.
The above discussion is the best available information of which I am aware on this problem - it should not, however, be viewed as authoritative as that from a textbook.
The reason that a model of elasticity is needed to calculate the moment of inertia is that the stresses in the hoop (or disk, or sphere) can contribute appreciably to the moment of inertia.
If you look at the plot of angular momentum of a rotating ring vs angular frequency in the above webpage, you will see that the angular momentum actually reaches a peak, implying that the moment of inertia has dropped to zero at a high enough angular velocity.
However, a further analysis shows that when this peak is reached, the model has made non-physical assumptions. Basically, such a model represents a case of a wire that is stronger than is physically possible.
See also the important section on the limitations of the model
http://gregegan.customer.netspace.net.au/SCIENCE/Rings/Rings.html#LIMITATIONS
The model does seem to be reasonably well behaved if one is careful to avoid assumptions that make the wire "too strong". In particular, it is necessary to use values of the constants such that the speed of sound in the wire is below the speed of light in the stretched wire as well as in the unstetched wire in order to get a well-behaved model.