We have the 12 pack Solar Bear, which is 14"x12"x7", well within American Airlines size guidelines for a personal item (18"x14"x8"). The Polar Bear backpack would also work at 12"x7"x18". You might even get away with the 24 can at 10"x18"x12".

I have put my passport in the cooler outside zip pocket, and put everything else in my roll-aboard. The only problem I have encountered is security in San Juan, who made me unpack the whole thing, which was neatly assembled like a three-dimensional jigsaw puzzle - the one time we were on separate tickets and tight on time! Otherwise you are fine once you have cleared security at your home airport.