When I was disciplining staff at work we had a formal process which asked the staff member to sign the minutes at the end. It meant nothing because if they refused to sign, I would simply note that <xyz> refused to sign. If there is an outcome that you disagree with then you need to appeal the decision. Refusing to sign (unless you think the minutes are inaccurate) is a redundant exercise.
Your manager needs to follow a reasonable process (I.e.: gives you a chance to defend yourself) and come to a reasonable (I.e.: rationally defensible) decision based on a reasonable investigation (I.e.: has actually sought and weighed up the available evidence).
If there is no CCTV evidence, then the only evidence available is your word vs the customers word, additionally the fact that the customer only paid once is evidence to support your version of events. Explain to your manager that you understand the customer may have been embarrassed because it looked like they had done something wrong, you didn’t think they had, and you were polite and professional at all times, however they were (understandably even if unnecessarily) embarrassed. Therefore the evidence available does not sustain a customer complaint because without any further evidence that you were rude, the reasonable interpretation of the facts as presented is that you weren’t rude.
The other thing you can do is contest the customer evidence: ask if the account the customer has given sounds like you, if you think the account the customer has given matches what other evidence is available (if they said you were angry and shouting, even if CCTV doesn’t record sound it would record aggressive or upset body language). The idea is that you are providing and interpretation of the evidence available so that your manager can make a decision in your favour.
If they have still made an unreasonable decision (that is, the evidence available does not sustain what they’ve decided) then appeal the decision to the next level of management.
Remember that in matters of employment the standard is not “beyond reasonable doubt” but “on the balance of probability”. You need to not only defend your actions but also provide an account that paints the customers complaint in such a way as to cast some doubt on their version of events.
eg: “Yes, I remember the interaction. I was monitoring the self-check out as the company is concerned that people may make mistakes and underpay or overpay in error. I noticed customer <abc> had attempted to pay via card tap and it had failed to go through. I followed them to advise them of the mistake and they reacted badly, believing I had accused them of stealing. I did not accuse them, nor did I believe they had done this deliberately, however they were understandably embarrassed. I believe this is the source of the complaint, I managed to get them to come back and pay, and at all times maintained a professional and polite demeanour. They were upset, however I believe that my professional duties obliged me to ensure that they paid for their shopping.
That the customer either a) made a mistake or b) attempted to steal is not in question- you can check the CCTV and bank records. They only paid once, and that was after I got them to come back in. I do not think they were attempting to steal but I believe I carried out my professional obligations in a polite and courteous manner, however the upset and embarrassment they have obviously felt now makes me believe that perhaps they were attempting to steal and were upset at having been caught out.”
This response gives a reasonable version of events that accounts for your behaviour, their behaviour, and if your manager is being reasonable would be very hard to sustain a customer complaint.