When a hotel does not publish their rates for their top suites online, I get excited :-)

Here is my strategy (tell me if I need to be more aggressive!)

On some occasions, I would book the highest suite type that can be booked via AMEX FHR (usually there would be no further upgrade as the next suite type would be the specialty/theme/historical/presidential types).

I would then contact the sales department at the hotel directly (cutting out AMEX, the travel agent). The word to use in my communications would be "upsell" as opposed to "upgrade". Most of the time they would come back with an "upsell charge" that is added to the rate of the original room type booked, while other times they would propose a higher room rate.

Note: AMEX Travel gets a 10% commission for every FHR stay they book. As such, I try to get the hotel to take another 10% off the "upsell" charge and have it show up as a separate line item on my folio (this way the hotel doesn't have to pay commission to AMEX Travel for the difference which they would have to if they just changed the room rate in the system). This kind of "back room" deal seems more easily done in places like Asia.

It helps to know what the "rack rates" for your original room type is as well as the rack rate for the specialty suite you are interested in (you can sometimes call the front desk anonymously asking for the "rack rate" for the **** Suite). The FHR rate is usually based on their "Best Available Unrestricted" rate which can for example be 70-90% of the rack rate depending on season. I would apply this percentage to the specialty suite I am interested in and use that as a gauge as to the maximum price I would pay would be.

Should the hotel come back with a rate that is more than "my maximum" price, I would try to get them down to this price or negotiate for things like extra Food & Beverage Credit, Airport transfers etc. Otherwise I would stick with my originally booked suite type and try my luck again at check-in (if it is for a short stay).