Welcome to FT gimme-gimme

Applying again isn't a problem, especially if you ask them to take your credit limit down for your existing card. They shouldn't pay you the sign-up bonus again, but if they do, then that is not your problem anyway - it is up to them to realise you have already had a bonus. Supplying your BD number should ensure that.

Either phone again and tell them you wish to apply and explain the circumstances, or just call and reduce your credit limit (thus making some available for the AMEX) and then apply for your new card.