I just realized I wrote myself into a corner, and the only way out was some very in-character hoop jumping on Bailey's part
Don't know if I can explain it well, but:
Bailey gets the armor, and elects to be a hero with her civvie ID a secret. She has family she wants to keep safe and far away from this danger. Niblings who deserve to grow up living normal lives and not having a crisis about being part space-alien and free of the responsibility that acknowledging their superpowers would give them.
The armor is reported lost, and likely stolen after the fight at the end of Takeoff (I'm working on it, I promise) wrecks the museum.
Bailey is now running around as Golden Eagle, helping old ladies cross the street, saving kittens from trees, and foiling the occasional Rogue's plot when she gets the chance. (You can't swing a cat without hitting a speedster in Central-Keystone. Not a lot of space to do the hero thing...)
Unfortunately for Bailey, she's forgotten one loose end. Nightwing had offered to help figure out this mystery, via a "contact" he was enlisting.
So now a new superhero shows up on the scene, in armor that seems awful close to Bailey's descriptions of the missing museum piece. Dick has seen connect-the-dots pictures on cereal boxes that were less obvious than this. But... There's always room for an incorrect assumption, as he's learned. And what would he even say? "Hi, I know your secret identity?" No! He has to be subtle about this.
Bailey is grinning like an idiot in the face of victory, when Nightwing melts out of the shadows like the weirdly attractive vampire he is and scares the daylights out of her. And then he's questioning her about the armor, where she got it, when she got it, does she know about the recent events in Keystone...
Bailey panics. I can't tell him the truth! The more people who know about Bailey Adler, the more danger my folks could be in. Besides, what if he thinks he has to reciprocate? That's way too much trust and responsibility, he doesn't know me, I don't know him, nope, no way, time to lie my ass off. Sort of.
The armor is hers. (Technically true!) She decided Earth was a decent place to live during the Thanagarian Invasion and decided to stay. (Also true! It just implies a Big Fucking Lie!) She's been quiet until now because... none of your business?? Why do you want to know her personal life?? She has to leave now for totally non-suspicious reasons that don't have anything to do with anything. Goodbye and good day, sir!
...Lying to him would be so much easier if her brain worked properly when he was around. Stupid charming jerk with his stupid pretty hair and stupid fucking smile... Ugh.
Dick, a little relieved by the fact that she has some kind of cover story and/or improv skills even if he doesn't understand where this sudden hostility came from, decides to try and have her defend from the other direction as well. He shows up at her apartment in Keystone with everything he can find on this new hero, the armor, Thanagar, and Andrew Fell.
Bailey tries to kick him out because she still can't stand Richard Fucking Grayson, but relents when he points out that he crossed half a continent on the request of their "mutual friend" -- not Wally, the other guy -- and gives him the space of a cup of coffee to make his elevator pitch.
And suddenly (because like hell is this jackass gonna figure out her secret identity) Bailey has to start insisting that there's no connection between Thanagar and her family, the necklace and the armor were a funky coincidence, such a shame it was stolen after the museum exploded -- no, no, it didn't look anything like the armor the lady in the photos is wearing, something was clearly lost in the game of telephone, oh look your coffee is empty time to get out of my house forever goodbye now!
...Dick conveniently "forgets" the file on Fell as she scoops up his papers and shoves him out the door. He does want to help, even if she doesn't want to let him, and damn it all... It's been awhile since he's gotten under someone's skin like this without actual violence being involved. She's... kinda fun to annoy...