If you're flying AA metal the whole way, youre SOL. Bout all you can do is skip LAX-MIA-CUN and buy a 1 way LAX-MIA or LAX-FLL. Upon arrival @ LAX you have to claim your bags and go thru customs and the recheck your bags for any connecting flights.(Had it been HKG-MIA-CUN it would have worked) So dont say anything and let them tag it thru to CUN
* the main problem with this is should your bag not make it to LAX with you AA will send it to CUN!
If youre gonna get a 1 way LAX-MIA just make sure its not on AA! Personally Id go B6(JetBlue) to FLL and get p/u there or Tri-Rail it down to MIA