Hello thanks for your patience.
Yes, absolutely, you have more than enough evidence. They have confirmed an error on their part and you have been messed around, told different things, and have relied on their word, so you have little choice but to take court action if they are now refusing to honour what they said.
What you would need to do is write them a formal “Letter Before Action”. In this letter you will need to state the amount you are seeking to recover and the reasons why. You will also need to give them a deadline to raise the payment by, usually it is 30 days in which to receive payment.
If payment is not received within 30 days, then you will need to issue a claim against them in the small claims court. You can do this by visiting this site https://www.gov.uk/make-money-claim
Or you can do it by post by using form N1 which you can find here Form N1: Make a claim against a person or organisation (Claim form CPR Part 7) - GOV.UK and posting it to:
County Court Money Claims Centre
PO Box 527
Due to the value of the claim you do not need a solicitor and can do this yourself. Once you issue the claim, interest will be added onto the amount from the day you issue until the date judgment is awarded. This is pursuant to section 69 of the County Court Act 1984. There will be an issue fee to pay, but you will recover this back.
It may very well be that the letter before action you send will make them pay promptly and you do not need to issue, but incase they don’t pay you, then the letter before action enables you to be able to start proceedings against them.
From what you have stated it is likely judgement will be granted in your favour and you will recover your money back. If they don’t pay after judgement has been awarded, it means you can then take steps through the court to enforce it against them.
Does this help?