Yes, you can book a night at the airport Hyatt through
DCL if you have not made your final payment for the cruise. I don't know if they would let you add if after your final payment has been made.
I don't think you really get anything kind of discount by booking through DCL more of the convenient of "one stop shopping" and luggage pickup in the room.
We did this last year and again this year because we are coming in with senior parents and they will be exhausted from the flight. They always have assistance from the gate (wheelchairs or the golf cart) so we have them dropped off right at the lobby. I go in with them to check in and DH goes to baggage claim for the suitcases. (We always come in the day before the cruise.)
On the morning of the cruise our luggage is picked up in our room (usually between 8:00 - 8:30) and delivered to the ship for us. That way DH and I only have to handle the folks, a day bag, and of course, Mom's purse.
It is very nice to stay there at the airport for the night if you are coming in late afternoon or evening, but a bit pricey. I think our cost this year per room was $210 for the night. But for us the convenience for Mom and Dad make it well worth it.