I thought I'd follow up with the case resolution.
When I opened Device Manager and chose to see hidden devices, a third, hidden tape library appeared. I have no idea where this hidden library came from, but maybe it's related to the firmware update on the library. As soon as I deleted this hidden device things were a lot better. Veeam saw the tape library with the correct number of drives and the correct number of slots.
When testing the tape backups 1 job out of ~20 failed with the error:
Loading tape 000650 from Slot 6 to Drive 2 (Tape2) Error: WinApi function 'DeviceIoControl(ChangerGetElementStatus)' failed, code '1162', message 'The indicated element does not exist'
No media in drive
This is an error we've always had and only happens with 1 or 2 jobs each week and the job that on which it occurs varies. Having sent the logs to support they confirmed that for some reason Veeam was still trying to use the 2nd tape library, rather than what I'd configured within the backup job settings. In order to fix this I opened Device Manager and disabled the library that I didn't want use. I also disabled the three tape drives that I didn't want to use. The result is that the only tape library and tape drives that are enabled within Device Manger are the ones I want to use. So far this has worked and there've been no further errors.