I'm going to add two things to make Sean biologically both of theirs. Sean was made half from Bruce's DNA and any gaps were filled with Lazarus Water. Danny also Ghost Adopted this kid and, when Sean accepted Danny as his dad, the Lazarus Water changed to suit Danny's DNA. Sean is, technically, a Halfa.
This, plus the very little information Bruce has on either of them, and Danny's off hand mention of parents experiments with Ectoplasm/Lazarus Water and threatening to tear him apart molecule by molecule? Bruce thinks that Danny was used as an experiment since he activated his Meta Gene by his parents and said parents wanted to create an even more dangerous super soldier, so made a child of Danny and Batman.
I'm also going to put that getting Ghost Adopted, or being in the Infinite Realms, paused Sean's age for five years. So when them pop up in this world, Danny's been Sean's parent for a good while. I'll also also add that Lazarus Water is highly contaminated Ectoplasm that Danny had to filter out with his own Core once he adopted Sean, replacing the contamination, slowly, with his own Ectoplasm. This is what makes Sean a Halfa. But, this made Danny really sick. So a good part of Sean's memory of Danny is being sick while also trying to train him in his powers.
The way Batman hears this makes it sound like Sean was kept away from Danny, his 'mother', until 11. The mad scientists (undeserving of being called parents, let alone grandparents) only let them be together when it seemed Danny was dying, had to be resuscitated multiple times, so that Danny would train Sean. Bruce thinks Sean and Danny were in a far worse situation than what actually occurred and both the little shits realize it. It's a good cover story.
Danny accidentally makes the whole situation worse for Bruce because he mentions a daughter named Eleanor and a son named Jordan that are no longer around (their clone bodies destabilized.) Mentions 4 others that died soon after they were "created" and one that died right as he was born (referring to the clones Vlad made.) Bruce is devastated at the thought of Danny going through that and even more at children of his he didn't even get to meet. He asks how Sean feels about his siblings and Danny just shrugs and says how Sean didn't even meet them, didn't know them and it comes out they weren't full siblings. Bruce is both relieved and horrified. So he asks who the other father was.
"My godfather. He was always such a creep, even before my parents found out I was... different."
Bruce is suddenly having an even more horrifying idea of what happened. Which gets worse when he finds out most of these other children were when Danny was 14 to 23 with all these deaths. Danny clarifies nothing cause he actually is a little stupid when it comes to anything other than mechanical engineering.
For more clarification , Danny is 35 and Bruce is 45. Bruce would have been 29 and thinks that, two years after Robin came on the scene, two mad scientist forced a 19 year old to have his kid.
If Dick finds out? He'd probably feel guilty. After all, it was probably because their was a child by Batman's side that gave these scientists the idea of the perfect child soldier. Especially since everyone thought Batman was Robin's bio father. They wanted a more powerful, Meta version of Robin to control and use.
Tim is originally the one that finds out where they live and is listening in through a bug he put on Bruce before the man left to talk to them. He also trying to figure out how to not get stabbed by this son, but it seems to not be an issue the second Sean realizes that it's Tim Drake. He gets so excited and starts rambling on about tech and how Tim is advancing the field of robotics and technology and how cool it is they're brothers. Tim gets overwhelmed and cries after the call.
This leads to Tim giving Jason the shovel talk.
"Look, his other dad is a mechanic. Who also makes guns. You can bond over cars and motorcycles, and even take him to one of those gun range things where you can practice together. But I swear, if you hurt him or make him feel unlovable or unworthy or even basic human decency, Gotham will keep finding your body."
Tim gives Damian a shovel talk too.
"Look. This kid isn't trying to replace you. No one is. You can expand a family without running out of room for more. He has a wonder other dad and probably isn't even going to live here unless on visitation or for holidays.
"He was trained to be a weapon, just like you, and he doesn't want to be one. You can probably ask about sparing together in hand to hand, but he's kinda a pacifist. But he has powers that could rival the Kryptonians. Treat him similar to how you treat Jon. I highly doubt this kid would ever be a Vigilante. But if he does want to join us eventually, you have to be okay with that. I also highly doubt that he'd ever want the title of Robin, since that's the reason he was created anyway and it is connected to a lot of his trauma.
"Try bonding over animals. You guys both love them. He even somehow befriended a glowing green dog, a giant horse made of shadows, three ghost Vultures, and I even saw a giant, fluffy Yeti like 5 times over at their place. They live next to a lot of farm land, so you can even have a look at those animals.
"Do not, and I mean Do Not hurt Sean. He already has so much trauma. All he wants is family. You don't have to like him. You don't even have to interact with him past basic politeness. But do not hurt him. On one hand, he's Bruce's biological son too, so you won't be let off the hook like when you attempted to kill me. On the other? Danny, his other dad, is very willing to kill anyone who touches his son and he already has. He probably killed his own parents to save his kid. And he terrifies Clark. I've never seen Clark so cautious of another person unless he can tell he'd be equally matched in a fight or would be defeated."