It’s exactly the same principle as in this thread - you need to create a separate “holding” bank account for Amex payments in QuickFile. For each day’s takings you are presumably raising a single invoice including the gross Amex amounts, so you will need to apply several partial payments to this invoice, one for the cash total (paid into petty cash), one for other cards (however you’re accounting for those), and one for Amex (into the new holding account). When you receive the net payment from Amex you tag that as a bank transfer from the holding account to your current account.
When you get a statement at the end of the month you then raise a purchase invoice for the fees and pay that from the holding account to get everything to balance.