That is a very interesting question. I don't think people between 18 and 21 are flagged on the manifest in any way, so they would have to see your ID to verify your age. Having said that, it is possible (though unlikely) that they've programmed the scanner at LAX to specifically check people's birth dates (that information is included in your booking).
I'd say the only way you can find out is to try. Or perhaps someone on FT going through LAX in the next few days can ask the lounge staff on your behalf.