What about DL LAX-FLL nightly red eye that is rarely ever cancelled/delayed... By virtue of the AF J ticket and w/ the red eye being considered the next day as per its arrival being the next day you should be able to use both the private check in for Delta One & then of course the D1 lounge.. Eat & drink a lot, pass out on the plane- if you drink enough you wont realize the A321-Neo has an insanely uncomfortable bottom.
FLL to MIA is a breeze and you can stop and have some Cuban coffee and pastries.
I know coming up to LA isnt ideal but just throwing the idea out there. if there is an issue with that flight you just crash at the Hyatt & take the 6am on AA to MIA.
Just an a thought...