I have spent quite a bit of time debugging this, and still can't figure it out.
While debugging, I ended up turning off the "Sync Books" option in iTunes. My iTunes sidebar, after clicking on the phone icon in the top left, would look like this,
except for one thing - every other time I would sync, the "Books" entry in the "Settings" section would be missing. (In order to make the "Books" entry appear or disappear, it was necessary to quit iTunes, and restart it - just leaving iTunes running and syncing again wouldn't do it.)
I figured out how to make this strange phenomenon stop - I opened the Book app on the phone. Even though "Sync Books" was turned off, I could see all my books. If I set the phone in aeroplane mode, though, I was unable to access them, so I guess they are in iCloud rather than stored locally.
Once I did this, the "Books" entry in the "Settings" section of the sidebar seemed to become permanent. However, there is still bizarre oscillating behaviour. Every other time I run iTunes, if I click on the phone icon, and then the "Books" entry in the "Settings" section, I see this on the right.
So no books at all. Every other time, I see this.
That is, the books are all there.
Checking the "Sync books" box launches me into yet another cycle. The first sync seems normal enough, but after that, every time iTunes starts, I get this pop-up window asking me to transfer purchased items from the phone to iTunes.
This window now comes up every time, despite choosing "transfer" the time before. During the sync, 73 books (out of 74) are transferred from the phone to the computer. But it, doesn't matter, the next time, I am asked again. There is still cyclical behaviour here - every other time, clicking on the "Books" entry in the "Settings" section of the sidebar results in zero books displayed on the right side at the beginning of the sync, and 73 books at the end of the sync. But the other times iTunes is started, there is one book displayed on the right side at the beginning of the sync (the one I was last reading), and 74 books at the end of the sync.
Opening the Books app on the phone, at various points in the cycle, does not seem to change behaviour. Clicking "Don't Transfer" launches me into another weird alternating cycle, which I can describe further if needed.
What could possibly be going on here? Has anyone seen anything like this, or know how to fix it?
In all cases, the oscillating behaviour occurs when the iTunes program is closed and started again. Just clicking on "Sync" again doesn't do it.
iPhone 6s plus - iOS 12.3.1 iTunes 126.96.36.199 on recent MacBook Pro