If you are referring the Hyatt downtown Miami then you are close to the Miami port. It will be about a 1/2 hour trip to FLL in the early morning.
A 6:15 flight will mean you have to get up around 2:30 a.m. (if you want to shower and not be overly rushed) in order to make it in time. Yuck.
I would go for a FLL hotel.
Also note that your trip from Bayside to FLL will cost you over $60. A limo is usually more expensive.