Just marketing... the rate booked via Hyatt only includes the complimentary night. You are right I can't get it to reflect the final price via hyatt.co
I just called the hotel to confirm this. The 4th night free promotion is based on a daily rate of $524 for a garden view room, which comes out to a daily average of $393. But as a GP Plat, I can get a room for a daily average of $382 per night.
I hate when hotels try to use clever marketing and calculus numbers for these promotions.