In Australia, I have not come across a 'Hot Plates' program offered by AMEX, only Fine Dining while I was a Plat and now as a Cent. So I can't offer any comments on 'Hot Plates'.

All I can say is next you try first with the restaurant direct and if unsuccessful then use 'Fine Dining'. Another method would be when ringing 'Fine Dining' ask the executive immediately whether another member has booked a table on the same day as you. Since this would be an internal allocation the person who answers the phone should be able to advise you without needing to call you back later.

All the best! Enjoy those fancy restaurants!