I've stayed at this hotel 20 times mostly leaving around midnight and have never been offered anything other than paying a full night's rate after 4-5pm (which as Plat is the free late checkout). I actually have never specifically asked.

At other hotels in town: Marriott was the most accommodating with sometimes allowing a 7pm checkout at no cost as a Gold/Plat (and would have done the same for others with a 50% room charge).

At these hotels, their housekeepers are usually finishing up for the night around 7-8pm. If the room is still occupied at that hour they cannot rent it again that night.

For the money involved, it may be better to check out and store your bags, go off to spend some money on a nice dinner, walk around the Pearl, or the souk, and then come back to collect bags and head to the airport (although note that the souk is on the way to the airport).