So basically BCA owned the vehicle as it was entered by their subsidiary We Buy Any Car and both appear to have failed to identify the mileage data discrepancy while carrying out their own due diligence checks.
The data discrepancy may only be an error but it is material to the description and therefore it can no longer be considered a UKCGR premium vehicle.
Under the circumstances,I would request ( by email ) that they rescind the contract but alternatively you might consider accepting the vehicle if the selling price was reduced by £1500 or whatever and the Assured fee cancelled.
You could also enquire from the BCA guy which section of BCA T & C’s you need to refer to with regard to mileage discrepancies effecting Assured entries with BCA owned and entered vehicles and their right to charge storage fees while they are investigating a problem which they themselves have caused and that you require a definitive reply within 48 hours.
I would keep emailing them and always be polite if they call you.
This appears to be straight forward ,there will be various other angles you could follow.
Please keep us informed.