Sorry, I don't think you could get daily rate under USD$300 in Park Hyatt Tokyo, even you use AMEX's FHR rate. Non-FHR rate is about USD$445 exclude tax and service charge
But, if you have AMEX Platinum or Centurion card, you may asking your PTS or CTS staff to book the following offer for you:
Stay 2 Nights and Receive a 3rd Night Complimentary at participating Park Hyatt Hotels
Valid: 06/01/2008 - 09/27/2008
Stay 2 nights and receive your 3rd night complimentary at participating Park Hyatt hotels. This is in addition to your Fine Hotels & Resorts amenities which are: room upgraded based on availability, daily continental breakfast for two, 4pm late checkout and a special amentity which varies by property. Please call your Travel Counselor and mention: E1320308 to take advantage of this offer.