No Mexicana lounge in ORD, BTW. There's a BA lounge and an EI lounge. Dunno if they are open that early or what they will think of your Mexicana ticket (probably not much) but you can try if they are open.

3 hours is way too much. If you can use the F check in line, 45-60 min will do just fine. I haven't flown MX since they bailed out on *A so I have no experience with using that F line w/AA status, but it moved fast w/*A status.