It wasn't checking FOR clearance, it was checking for denial. Cara wouldn't pass because she's a registered republic commando. the assassin wouldn't pass because she's wanted. fett wouldn't pass because clone.
mayfeld specifically states in the episode that the console checks databases to deny people. it doesn't check lists to approve them.
Mando never worked as a storm trooper or anything.
If that's true that might be the dumbest thing written since the Sequel movies. You really think you'd have a security system that doesn't check to see if you're part of the organization it belongs to, then your clearance? Nope let's just see if a live person is staring into it. That makes zero sense.