Have you got the original battery terminal clamps. If it's 12.whatever at idle it may not be alternator fault completely. Try take off all wires at alternator and clean and put back. Clean negative at body...clean all negative.
So - I've confirmed voltage at idle is 12-12.6 or so. When revved, this goes up to around 13-14. This likely to be contributary?
Really not looking forward to replacing the alternator if required - I understand this is a rad out job!