After all the standard checks, I found a support thread that suggested that when Apple updates the IOS, it can confuse the existing download of iTunes – even if it’s the most current version of iTunes.
Solution: Their solution was to uninstall iTunes and reinstall it.
I had the patron uninstall and reinstall iTunes, and it worked! All her existing iTunes content was still there. (She didn’t lose it when she uninstalled.) When she re-transferred the book, it showed up perfectly.