Unfortunately it is only the MY2020 and onwards that the NX was fitted with Lexus Link and therefore logs every journey the vehicle makes.
Without this, a dash cam or tracker it is essentially impossible to prove the extra mileage occurred, other than a photo of the odometer when you dropped off the vehicle.
I find the attitude of the manager disappointing. I'd understand it if this was a small independent backstreet garage where I'm sure they get the occasional person trying it on, but an expensive vehicle being serviced at a franchised dealer - I'd expect them to take your word on the matter.
Personally I'd make sure no one you know ever uses them again.
I'd suggest complaining to Lexus GB, but I have a strong suspicion that they will just side with the dealer, which they seem to do 99% of the time 😞