First of all, I am a shareholder in AIB, in case anyone thinks this influences my views.
Secondly, it's completely unacceptable that when an issue creates concern that they cannot manage to provide a phone number where people can get information immediately without having to hold.
Now to the core issue...
Forget the fact that it was a mortgage for the moment. Let's say it was your ESB bill. They would write to you saying that you had been undercharged by €1200 and please pay up. You would have had the benefit of the money for the past 12 months so you gained from their mistake. They would not give you a long term loan, but they probably would give you 12 months to pay it.
Now look at the actual case. It is a mortgage. You have benefited from their mistake in that you have had more money. You owe them €1,200. They are not profiting from your mistake at all. They have lost out.
What if it was an overcharge? AIB refunded all their overcharges with interest. Presumably they are not charging you interest to date on the undercharge?
Let's be clear about this. Everyone of us makes mistakes. Every company we are employed by makes mistakes. If a customer forgets to invoice me for a few months, should I refuse to pay it?
I think Buttermilk's suggestion is a fair compromise:
Tell them you want a 0% loan over 12 months and will repay it in instalments of €100 per month or you will take the matter to the relevant authorities.
And, if you are still not happy, you can do as ClubMan says and go to the Ombudsman who will offer a completely independent assessment of the issue.
Brendan