Most unlikely since the cards are ssued by two different organisations, the bmi AmEx by MBNA (part of Bank of America) under licence from AmEx and the BA card by American Express itself.

Bonus miles for new BA cards may be 'miserable' in your view, but better than the 0 miles offered for a new MBNA card to somebody who has received opening spend bonus miles before. (At least, that's what the T+Cs say.)