I could find no mention of any charge to home buyers for using the NHBC Resoloution service.
There used to be a £50 deposit to guard against spurious claims brought by buyers and designed just to waste everyone's time. This was refundable if the complaint was considered justified.
You will need to read your NHBC policy documents.
In any event it was Barratt that have called for this and all charges/costs (if any) should be met by them.
You should be aware that Barratt have a very close relationship with the NHBC; exclusively using their warranty and in most cases, the NHBC also carry out Building Control inspections.
Barratt have also won more NHBC Quality Awards than any other house builder since 2008.
As such I would be surprised if the outcome will have been agreed between the NHBC inspector and Barratt's representative before they even ring your bell!
Should you not accept the NHBC inspectors resolution report, you could go to arbitration but would then be liable for the arbitrator's fees and other costs.
My advice would be to bombared Mark Clare with e mails and post regularly on Twitter adding @Barrattplc in each "tweet"
You could also join the Facebook Barratt buyers action group:
https://www.facebook.com/groups/230629503676904/