Does a chapter published in a book fit as part of a paper-based dissertation?
While regulations may differ from place to place, I find it highly unsurprising that a book chapter would not "count" for a paper-based dissertation.  The reason is that book chapters are typically peer reviewed lightly, if at all.

As such, if your dissertation requirement is effectively "N peer-reviewed papers," then this book chapter is highly unlikely to be peer-reviewed in a manner consistent with the expectations of the university.  Just in the same way, your professors would probably not count a paper in a predatory journal as helping to meet your dissertation requirements.

Note, however, that this doesn't mean you should not write the chapter, if it is a good publisher and a work you would like to create.  Book chapters can be a good way to organize your thoughts and put together reviews, and in some cases can actually be quite highly cited.  Likewise, the text from the chapter may turn out to be useful as part of your dissertation (depending on university regulations about copyright and theses, of course).  It's just that it's probably not a peer-reviewed publication and will thus likely not count toward the number of such that you need for your dissertation.