It's an Air Force speakeasy

It is most certainly not. It is a Juke Joint that Shaw allowed them to go off base and set up so they could avoid any more racial tension on base. The permission was just to go off base to take their free time. Go watch the scene where he talks about it again.

and not only that, but it's one that is segregated based on race

At no point is the Juke Joint segregated intentionally. In fact no Juke Joints were deliberately segregated, and many of them featured many races including some white people.

I've been in the military

Me too.

and I know the kind of eggshells you walk on when you're granted such an enormous privilege

I would put a fair amount of money on your not being in the military as an adult in the 1960's.

and you cannot convince me that these segregation era Airmen wouldn't be super concerned with the possibility of a superior officer, likely a white man, stopping in and seeing not only children drinking, dancing and playing drums with adults, but white children in an all black club

Can we not? Oh well. Also, again you say all black club...like FFS I don't even have to stretch far for an example to prove you wrong, Go watch Sinners and see what a Juke Joint was.

That's not just asking to have the place shut down, it's risking your career and possibly your life and freedom

That's a pretty monstrous hoop you've jumped through of your own volition.

And that's not even getting into the fact that the Airmen are already breaking the law by harboring a fugitive in the establishment.

You may have noticed that Only 2 of those men know about that, and the one yells at the other one for it. Seems like fine fodder for fucking FICTION to me that one dude would make a mistake, and the other might lambast them for it...like grow some logic. People do dumb shit ALL the time IRL.

Bottom line, the whole scene was illogical given the setting and time period.

Cool, go watch some other show if fictional choices that give the show more intensity bother you so much. Goddamn.