I use Hyatt.com rates almost exclusively for all my Hyatt stays, and pay with a Mastercard.
Every single one has counted for stay credit, GP base points, GP bonus points, and FFN credit. When I check out, I make sure it has the "Mastercard" notation on the folio. It has only been missing once, and I had them add it then and there.
My rate code on the room key holder always reflects that I am using a Hyatt.com rate. So not sure why some hotels are having a problem with this.