Bah! I don't see a way around this other than a manual export and import like I described. Maybe somebody else will chime in with a better idea.
Ok, I fixed it.
What I somehow missed was iCloud Groups etc occur straight after the On My Mac list. By deleting all the On My Mac cards plus all the groups, the iCloud Groups came into view, and all the Birthday duplicates are correct.
I have no idea how I could have missed the second list of Groups, unless it only synced/appeaded overnight
But all fixed!