AIBU for refusing to sign a letter of improvement at work? Retail customer service related

[Welshlady60]

Recently at work I was made aware by my manager that a customer had complained about me to HR for my apparent poor customer service towards him.
In a nutshell, the customer had used the self service till, but his payment of around £35 hadn't gone through.
I went outside and asked him to come back into the store, explaining why. He wasn't happy as he was just about to get into his car, and he insisted that he'd paid as he 'heard the machine beeping'.

Once again, I explained that the payment hadn't been taken, and so he came back into the store, angrily tapped his card on the pad and stormed off saying that he 'wasn't trying to steal'.
I hadn't accused him of anything, and was polite at all times.

He's now made a complaint against me, which I wouldn't normally be bothered about. However, I was asked to make a statement to give my version of events, and was told that HR may most likely decide to issue me with a letter of improvement.
I've been at the store for ten years and have never had any issues, but I'm annoyed that a customer's unjustified complaint can result in me being reprimanded for doing my job, especially as I wasn't rude, he was.
Should I refuse to sign it, given that I feel it's an unfair complaint?

If the staff member had followed the thief outside then they could've ended up being suspended (depending on the stores policy) as the store only usually insures their staff if they're attacked, for instance , inside the premises

The shoplifters know that they must likely won't be challenged once they get through those doors, unless there are security people around.

The manager is definitely aware that we ask people to come back in if we see them for payment, she's done it herself if she happens to be helping cover on the till in busy periods,

Thanks for all the replies.

The manager was away at meetings yesterday (and will be today) , I had a quick look through the stores policy and couldn't find anything on what to do with a failed payment
It only mentions not going out after shoplifters (what I could see anyway) , and other staff members weren't sure what we're supposed to do in that scenario either, regarding the self check outs.

I may well be told that I shouldn't have gone outside because the customer could've got aggressive with me and attacked me, then technically I'm not insured,

But there's no clear guidance on this, as we don't class people walking away from the self checkouts as 'shoplifters', as yes there is a lot of theft on those machines, but I think it's usually from people under scanning all their items.
Many people walk out not realising their card may have declined for example, or there could be an issue with the machine not processing payments properly.

Why bother having to keep an eye out on the machines if we are supposed to just not do anything about it once the customer exits the store? I've already mentioned there's a bit of a delay telling us the payment has completed after the customer taps their card.
Those machines are a pain the arse!