Dealer is being a lazy arse, or just incompetent, just had a friend do it and she took proofs into the dealer and he sent them off to DVLA and she picked up the car with her free disabled class taxation done.
Tell the rep to pull his finger out and check what is his responsibility, take your evidence for them to copy(or just take copies) and refuse to sign for the car until it’s sorted, once the dealer principal finds out a sale isn’t going through because one of his reps can’t be bothered then things will get done.