I hope GWR's response does indeed represent an advance towards a satisfactory outcome, but I'm completely perplexed by the terms in which that response is written!
“This case was originally in court and was resulted (!)... The provider … was not notified...” How easily the passive voice can be used to obfuscate and potentially draw a veil over somebody's error.
From the sound of it GWR brought a successful prosecution that led to creation of a DBS record for the defendant – who still appears, from the reference to “your record”, to have been your stepdaughter. Someone unknown made an application for the case to be re-opened and someone – probably GWR – then arranged for the charge to be withdrawn. Who is this “provider of our Arrest Summons Numbers that creates a record to show on DBS/CRB checks”? Isn't this someone within GWR Prosecutions Department – why would GWR outsource this task to some external agency? If this is just a circuitous way of saying that someone in GWR didn't become aware of a case that company had itself withdrawn then GWR are at fault for having failed to prevent an erroneous record finding its way onto the DBS database. They, after all, are the ones who initiated the prosecution and presumably agreed to its withdrawal.
Alternatively, if the court failed to notify some external agency of the case's withdrawal when it should have done so and this has led to the creation of an erroneous record then it's the court's error.
In either case, a fresh DBS check seems to be required at a further fee of £44.00 for an enhanced check, and it seems clear that this is a fee the person/company at fault ought to bear rather than your stepdaughter.