Thanks for the reply. I find it odd that the Books apps on the Mac and iOS devices all have "Books", "Audiobooks" and "PDF" tabs, but that that books and PDFs sync between devices, while audiobooks don't. Not Apple's most elegant implementation.
So I synced the audiobook to my iPad via iCloud Drive and when I select it it opens the Music app and plays. However, I can't see it anywhere in the Music app. There's no "Audiobooks" tab here. If I exit Music, then the only way I can find to restart playback is to go to Files and reopen the audiobook, which starts it playing again from the beginning. I have no idea how to pick up where I left off.
Oh, wait. I can open the file in GoodReader and I can quit and reboot that app, reopen the file and it seems to remember where I left off. This is workable, but not as elegant as listening in Books. I'm unsure if GoodReader's sync features (I've barely used the app) would sync the latest location between devices, but I can at least listen when I'm away from home, so thanks.