FHR (as well as the Citi 4th night free booked by Carlson Wagonlit) certainly got me the SPG and Hyatt credit as expected, so it would be good to know if Hilton as a chain specifically does not recognise such bookings for points or benefits.

FHR rates are GDS rates published by the hotel, and not pre-paid or specifically discounted under a separate negotiation by the hotel. FHR is simply acting as a travel agent designed to attract high value customers by providing pre-agreed benefits, which is why other chains always give points and status benefits for such bookings.

However, non-FHR bookings on the Amex website is treated the same way as Expedia or Booking.com etc, and do not earn points or entitle program benefits.