I completely get why the Hyatt is/may be valuable to you but it doesn't mean you "need" it, and I think you are better suited by waiting. It's not the end of the world and will help you in the long run.
Of course, you can feel free to ignore this input, but there's not much point in getting approved for the Hyatt if it means other creditors get nervous and shut off your accounts because of the aggressive credit seeking. Not trying to scare you, but it does happen. I would absolutely avoid apps for awhile but that's just me.
A good credit score is the means, not the ends as I see it and I don't mind having the score plunge for a while in order to get the concrete benefits of new cards.
But getting previous accounts completely shut down would be something I'd be worried about.
What are some data points for such an extreme response by banks? Especially for people with no baddies whatsoever and a decent income in the $100-$200K range.
From nearly everything I read, getting more credit actually improves the perception by other lenders due to lowering the util %.