Thanks everyone for the advice - I ended up finding some success in the end. Given that I was travelling with 4 people, I really wanted to find a way to make this work, and on the day of my reservation, I had found the same price on both Hyatt.com (which was not responding to my attempted reservation quickly), and Booking.com, so I went with the latter.
I responded to their e-mail, highlighting that 2 weeks had passed and I had also documented exchanges with Booking.com and Hyatt to confirm that my reservation was indeed in good standing. I countered with booking.com saying that I found a room for $3400 and change on their website, although with a Pool View, rather than an Ocean View. Given that this amounted to a $700 difference, I recommended a reduction in price by $700 and a downgrade to a Pool View room and after a few hours they agreed. So I am now confirmed with an $1800USD stay. In the end I am pretty happy with how this all turned out.