My initial reaction: no way. Upon reflection, I'd say that it's possible but highly unadvisable. If the FLL flight arrives at Terminal 5 (I think it could arrive at Terminal 6, which DL uses for some flights), it's probably a good 5-10 minute walk from aircraft door to terminal roadway. Then it's a solid 10 minute walk past Terminal 4 to TBIT, longer if you take a bus. Security waits vary widely at TBIT, even if you're not doing checked-baggage screening, which your cousin won't be thanks to through-checked baggage. You could breeze through during the day, but you might also have a 10-minute wait or a 30-minute wait, or a 60-minute nightmare. During the day, more likely to be a short wait, but I've arrived mid-day and seen massive lines snaking out of the terminal. Then you've got to get to the aircraft: 5 minutes if it's at a contact gate near the center of the terminal, a whole lot more if at a remote stand.

My own personal MCT domestic>>international at LAX is 2 hours, but three is preferred, and 4 if on separate tix.

I'd try to get on a later KE flight if possible, or suck up the connection through ATL if it would get your cousin into LAX earlier.