I just staggered onto Twitter, and managed to connect with them again. They also insist they can't close the account while there is ANY money in it. So they sent me info on how to re-register for online banking and told me to transfer the funds out and then it could be closed. I got all the way to the last step and clicked "enter" and it came back with "the system is unavailable at this time". So sorry.
I also learned from the subsequent Twitter agent that because this is the only account I have with RBS, I CANNOT close it online. It has to be done via letter or in person. So I have recopied the letter I sent last year, which was a variation of the one from the prior year. In this one I listed the bank info of a charity that I found online and have asked that any funds remaining in the account be transferred to that entity. The Twitter agent assures me that this can be done, and then the account will be closed.
Considering phone banking told me I could only close the account via online banking, and the website says it can be done by letter (three sent so far), and none of that has panned out, I am willing to try one more time via letter. I will probably also simultaneously send a complaint to their "Customer Relations" manager in Birmingham.
Seriously, this has become a real pain in the ars*.
I also asked if they could confirm the balance for me, and they said they couldn't unless I was able to log into online banking.
Of freaking course.
[EDIT: I was finally able to get into online banking. There's 36 pence in the account. AND there is
no way to transfer funds out of that type of account in online banking or to close the account online. Big "NO YOU CAN'T FROM THIS TYPE OF ACCOUNT" banner lights up on the page when I try to transfer the 35p. A somewhat snarky letter is going out to RBS Customer Service in Birmingham in the morning. I don't expect to hear anything from them, either. But it's due diligence.]