Found this document online from 2013.
https://www.bmwbrochures.co.uk/AUC_S...dedb34c881.pdf
Criteria may have changed over the last 10 years, though.
Interestingly, back then, they didn't need to be main dealer service history but only needs to have used BMW recommended parts and oils. I suspect most dealers don't go this route for the reason you're experiencing.
On that basis the car must have fully documented and verifiable service history to be a AUC.
Unless they come back with documentation for the service then you've been mis-sold the car.
Ultimately, there are a few options.
Reject the car and agree a refund that the car would sell for now as a AUC so you only end up paying the deprecation during the ownership.
Reject the car and ask for an equivalent replacement.
Keep the car and agree a partial refund to replicate the reduced value associated with the missing service.
Personally, I'd go with option one or two.