I've once had a pleasure of trying to convince a Swiss bank that I am neither US citizen nor resident. Fortunately it boiled down to a few phone calls and a couple of letter saying that I am really really not a US citizen, I swear.

This is why I suspect such an attempt would require going via the courts. You would say to the bank "I have unilaterally renounced my citizenship", the bank would not accept it, and you would be asking the court to order the bank to recognize that renunciation.

Why not tell them you were never a US citizen and present them with a passport from another country?

One catch is that if the passport is like the UK or US passports (i.e. perhaps all?) it will show your place of birth, and if (like the UK Prime Minister) that was within the US, anyone with half a brain will know that from the US perspective, you were born a US citizen.