Sorry I was referring to the Amex agents who tell you that you can't do this if you ring up to downgrade instead of using the link, not the BA agents you might get if you ring up to do a booking.
I think BA could tell which type of Amex card it is in the computer if they cared, but they don't.
There is a six digit IIN at the beginning of the card number. The first two digits identify it as an Amex card, the other 4 identify the sub-type and country.
(The whole number also includes a Luhn algorithm checksum at the end which stops single digit errors being accidentally typed in)
https://en.wikipedia.org/wiki/Payment_card_number.
Try typing the first six digits of your credit card number in here...
https://bincheck.io/