I think that there is no opportunity to transfer Amex MR points to Hyatt.
But what's about the vast array of British Airways flights to the various Hyatt destinations? Flight rewards of the Executive Club of BA (how their frequent flyer scheme is called) are very competitive if you compare them with Miles & More.
Therefore if I would be you I would change the strategy: Pay the flights to your favourite Hyatt destination with Mr rewards -> BA Avios.
Hope this helps.
Seems to me (based of the profile) that the Op has plenty of Air Miles on BA/LH available to fly. I can understand that he sort of prefers to book hotels.