Wolf genes being dominant, bird ones recessive, so over time bird dynamics are almost completely wiped out of the population. Bruce is an anomaly, both his lineages still carried relatively strong bird genes, so he is a bird, and takes in only wolf children oh my this is simply the cutest.
Talia was a wolf, so Damian is a wolf too, and Bruce doesn't even understand why he's sad that Damian acts like his other children, he doesn't understand his own instincts as a bird.
The kids don't understand him either. They don't understand why he's so strange, why he acts completely different to any omega they've ever met.
Then, as you said, another batfamily. Both sides are hostile, until it's established neither side has bad intentions, then they relax and start planning it out. Both Bruce's move to the batcomputer, and it's barely a minute before Tim is shoving under B's (AU Bruce) arm, getting tucked under his chin instead. Damian migrates over, and disappears under the cape, quickly followed by Cass. Bruce is staring, there's a craving in his chest he doesn't understand, his mouth is dry, and jealousy burning through him so intense he just leaves. Stalks out without a word, intent on getting himself under control. Everyone watches him go, Batkids confused, the Bats analysing, suspicious, more so when no one goes after him.
The Bats are invited to stay in the manor, and keep catching Bruce watching them from afar, then disappearing, always leaving a bitter scent trail in his wake. The chittering, the chirping, all of the unusual noises the Bats slip into confuse the family. The manor is full of distant calling, they can communicate at such lengthy distances, the family is fascinated. Then au Damian falls, and a sharp call has Bruce at his side instantly, making soothing chirps back, gathering him close. Damian leans into his easily, accepting the comfort, until Dick "Bruce?" has him freezing, and he deposits him into the newly arrived B's arms, vanishing.
It pretty much solidifies for the Bats what's happening, and they're beyond peeved. Damian is wandering the manor, unable to sleep, and hears the Bats arguing late at night. That's how he figures out Bruce is a bird, because he hears au Dick yell at B that the family couldn't care less about Bruce's biology. He isolates himself on a research binge, diving into the limited historical records on it.
I also want to deviate a lil bit from your og plan, dear asker, and I want au Dami to hear Damian trying to chirp. He hears the bastardised attempt, and lingers, listening to Damian's growing frustration with himself, until he hears something shatter alongside a yell, and enters. Offers to teach him. He's at an advantage, he has recessive bird genes, somewhere in his instincts, he knows how to do this, and spends days helping him perfect his chirp. Both families are curious what the Damians are doing, because they barely emerge from his room.
Then Damian seeks out Bruce, and, as you said, chirps at him. By the time he's cradled in Bruce's lap, he's seen the borderline terror, the instict-deep confusion, the shame, and wants to cry, wants to fight, wants to shout at his entire family.
Bruce, once Damian reassures him he wants to be scented, finds himself entirely lost, he doesn't know what to do. He's suppressed his instincts for so long, he barely learnt how to indulge them in the first place, he doesn't know what to do.
So he reaches deep into his instincts, and calls. A sharp, loud, panicked sound, that rankles Damian even as a wolf. Jason practically busts down the door, Cass leaps over him, chirping back, ready to help. Bruce, clutching Damian, looks at them, so, so ashamed of himself, and confesses "I don't know what to do."
The rest of the au family is gathered in a matter of minutes, until B is there, and his kids gather under him to reassure themselves after hearing such a distressed call from a Bruce. And he agrees to make a massive document for Bruce, to explain how to deal with his instincts, and act on them.
Okay I'm stopping there but this is definitely going into the list of aus I'm definitely writing, thank youuuu.