There used to be surcharges for paying with personal cards on BA,but these are long gone.

My guess is that your Amex has been (wrongly) flagged as a corporate one hence the surcharge.

You could wait and it might get corrected, or raise a dispute with Amex ,that would cut down on the waiting time.

Current rules for corporate card costs here
https://ndc.ba.com/faq/commercial/payments?bk=Y3