OK I have booked this hotel today through Virtuoso. I got all the amenities and the 4th night free. And instead of the long periods of time I spent on the phone to Amex and e-mailing Amex, I had a pleasant five minute phone conversation with the Virtuoso travel agent and had written confirmation of the booking shortly after that.
I wonder about the economics of this. The travel agent who booked this for me makes his commission. By refusing me the FHR deal, Amex lost the commission, but more importantly, next time I book travel I will look to my Virtuoso travel agent first as he got me the good deal without any hassle.
So far this Amex card is batting 50% for me, I will see how things play out over the next year before it is time to renew the card.