Yes, I have tried with my other ID but nothing changed. Anyway I'm sure of the ID I use to purchase books. Always the same one. I've been doing this for years.
Also, if the book was linked to my other ID, I wouldn't be able to read the book on my iPad that is logged in using the Apple ID that doesn't give me access on the Mac. And all my books (purchased previously to the recent changes in iTunes/iBooks) work fine, can be deleted and redownloaded and read. If I switch to the other ID, all my books disappear from iBooks. The readable ones and the unreadable one. I also checked the presence of the troublesome book in the Finder where it can be found deeply nested in the Library folder. It's there, disappears when deleted and reappears when downloaded again.
I also contacted Apple support (by chat). The support person suggested I repair permissions with Disk Utility and delete the preference plist. I did all this and nothing changed. The problem was escalated to another support person and later I was called by phone by yet another (senior) tech who got remote access to my machine (interesting experience). All to no avail. I deleted the book, redownloaded it many times. It's just round and round: enter your password, this computer is already authorised, the Apple ID you're using doesn't match the one that was used to purchase this book.
The tech support person who tried to help using remote access said he'll get back to me next Tuesday. If he finds a solution before anybody here does, I'll update this thread.
I must say I'm impressed by Apple's support. I didn't wait more than a minute or two for each session with a support person. They were all very patient and helpful and didn't act as if I didn't understand what I was doing, something I often notice with other vendors. The fact that the problem was not fixed immediately doesn't surprise me. I usually fix most problems myself, with or without help from the internet community, including this one. When I find nothing, as is the case here, it's usually something really weird, some obscure bug or some rare incompatibility. Over 30 years of using a Mac, this has happened a number of times.
Ellen
