If you ever do find yourself in the situation where the amount you wish to spend exceeds the available credit limit, then there is a simple solution which I have used several times.

Pay the extra in advance online using a debit card linked to your bank current account. This amount will be added to the available credit immediately.

For example, if you have a credit limit of £3,000 with zero current balance, but wish to purchase a BA itinerary at £3,800, then go online to your BA Amex account, select debit card payment, enter the details (for an ~£800 payment), then refresh the account page to check the 'available credit' has changed to £3,800.

Now purchase your itinerary and you will owe Amex £3,000.