I'm pretty sure only one person has to check in for the busses. I think my wife grabbed a couple of chairs in the lobby while I checked us in. I agree that there's not much time between when you check in for the bus and when you leave. In addition, you have to have your luggage outside the hotel room door by 9:00am, I think, so you will probably have checked out of the Hyatt before the bus check-in begins.
We always pack and leave our bags outside the door, then go grab breakfast before going to the bus. We like the little bagel shop in the terminal.