The only way to solve this that I could find was the following:
I had to get the original iPad that was used to make the backup. It had been wiped clean and reconfigured for another user. First I backed up its new data to another iMac and then wiped it.
Then I restored the backup from my Macbook to the original iPad.
Next I wiped the new iPad and then brought the old and new iPads together and transferred the settings and data between them.
Finally I wiped the old iPad and restored its new settings from the iMac.
In other words, I had to do what I did when I first received the new iPad. I was fortunate that I could still access the original iPad which was associated with the backup.
Can anyone confirm that its impossible to restore a backup from a Mac to a device other than the one associated with the backup? It makes no sense that this is the case, but I couldn't find any other way around it. Is this only when transferring from a Mac or PC? Does it work when backing up from iCloud - i.e. the backup can be restored on any compatible device?