Here's an update: success! Converted the Shakespeare in Calibre, the only setting I changed from the default was to set output format to epub 3. The conversion took a LOOOOONG time. This book has thousands and thousands of internal links to footnotes, as well as many chapter, scene, and other divisions. After conversion, the .epub file went from 36MB to 11MB, and the internal XHTML files were split into many smaller ones. I didn't see any file larger than 100K after the split, and most were in the 10-50K range. After sideloading to Clara, the book was immediately more snappy and responsive on page turns, especially when clicking the links to the footnotes (which usually have to jump to a notes page that's a long way away in the book, then return back to your reading position when done). Calibre handled this large conversion really well, I have yet to see any issues in the rendered output even after it did all that file splitting and link updating.

For sure, that's what made the difference in this case.