I called My Hyatt concierge to apply a SUA to the booking. She confirmed that the booking will earn EQN and points.
Someone upthread mentioned that all Amex FHR & THC bookings are made through Expedia. There must be some exception where FHR & THC bookings made through Expedia are treated in the same way as Hyatt-direct bookings.
From an Amex business standpoint, it wouldn't make sense for some bookings made through Amex Travel to qualify for Hyatt credit while having other bookings not qualify for Hyatt credit.
If anyone has recent examples of THC bookings not getting credited, let us know.