I've done this a number of times. I never bought a proper connecting ticket, fyi, done this about 5-6x before. Every single time I did it via LAX, where Southwest Airlines has flights to LAS every 1-2 hours at the longest.

Land at TBIT, bolt through, either wait for the awful bus or take the ~10 minute walk to T1, buy ticket on the spot. They sell something called "Business Select" which is regular economy (pretty sure Southwest doesn't have anything else?), but these types of tickets seem to be always available and let you board first and whatnot. The best thing about this is you don't have to worry if your CX flight is early or late, you just go buy your ticket on the spot. I have never once encountered a problem where these "Select" tickets are sold out, ever. I have Global Entry and from immigration counter to boarding my Southwest flight, I think I've been as quick as ~30 minutes.

The downside is if you're sensitive to spending an extra $200-$300, which is roughly the cost of the ticket. But I think it's money well spent if you're in a rush or don't want any hassle. Don't even peek at the schedules until you pop out of TBIT.

To hell with bothering with attaching AA flights or whatnot.