It's important to bear in mind that whether you're paying in local currency and getting Amex to do the FX, or paying on the phone with BA, that somewhere along the line there is going to be an FX discrepancy.

(What I mean is, BA will be using an internal exchange rate to charge you - this might or not be better than the Amex exchange rate inc. the 3%)

I don't know how much this difference would be but I'd expect it's worth double-checking before one assumes the 3% Amex charge is automatically terrible.