You should definitely be able to to stay until 4pm. I've often checked out at 4pm at various Hyatt properties, and then they allowed me to hang out in the lounge until I had to leave for the airport.

2am is pretty late. If it were just a little later, I'd almost try an early check-in for the next day. Perhaps if you contacted the hotel direct, you might see if you can stay in your room for the entire day, and only pay for one day.