I don't think that it is impossible for Mills theory to explain low orbit electron capturing. He does indicate how to treat this case in his latest versíon of the book and it does so by using the traped photon(s) that is indeed in contact with the nucleus and will when it colappses into the nucleus drag the charges with it. This is basically what you shohuld have in the QM representation as well, the atom sized wavefunction of the electron needs to collapse into the nucleus as well.
At face value, this description of inner shell electron capture is "not even wrong," as they say in physics. Perhaps you're not summarizing Mills's position adequately. Can you point us to the section that you're summarizing so that we can take a closer look at it?
In QM the matter is rather straightforward: the electron (especially s-orbital electrons) occupies a three-dimensional orbital that overlaps with the nuclear volume.