This feels like a good conceptual foundation for wholesome crack of the Good Yet Overwhelmed Single Parent Bruce Wayne Means Well variety.
For instance: It takes Bruce longer than the average person would expect to notice Tim, because he's a little face-blind and already tied to the idea of adopting orphans who look like him after discovering the way Dick's convenient resemblance ensured that even Bruce would recognize him anywhere. It also ensured that Bruce would guess more correctly than not which child to pick up from school, because even he's aware that "sorry, I thought that one was mine" isn't the best defense against kidnapping charges.
The point is, the resemblance proved so helpful with Dick that Bruce had no qualms at all in adopting Jason.
So it's understandable that, at first, he assumes Tim and Jason are the same person. They both look like him. Only very small. After years of living and working with Dick, and then Jason, Bruce has good reason to assume that any small black-haired blue-eyed child in the Batcave is currently Jason, especially when the child always answers to "Jason."
That should be reasonable proof that the child is Jason.
So even when he starts to suspect there are two children living in his home who look suspiciously like him, both of whom answer to "Jason," he's going to bury that suspicion so deep. Because it's taken him this long to get used to Robin's new name once already, and there was only one of him then. Now that Batman and Brucie Wayne are living full lives, Bruce barely has time and energy for breakfast, much less mentally reframing Robin as two entirely separate children.
Three, if you count Nightwing, which Bruce tries not to, because it gives him a headache.
Eventually, he sees Tim and Jason in the same place at the same time, shattering the fragile illusion and undeniably confirming that he has, at some point, inadvertently acquired an extra child without noticing. But by now, he's so used to calling his son “Jason,” and assuming there's only one of him, that he continues calling them Jason.
In his defense, neither of them seems inclined to object.
He eventually learns that Tim's name is Tim years later, minutes before the doors open for Wayne Enterprises' presentation at a green energy conference, because Tim is wearing a sticker that says "Hi! My name is TIM"
"Is it really?" Bruce asks, staring at the sticker on Jason Tim's lapel.
"Yes?" Honestly, it's become so normal for Bruce to call Tim "Jason" that even Tim hasn't thought about it much in years.
"Has it always been Tim?"
Bruce sighs. It’s taken so long to learn to call Robin "Damian" once Jason went and renamed himself Red, and now he has to start all over with learning to call a Jason "Tim." He will, of course, because this is his son, and he supports his son in all things, including learning to call him by an entirely new name.
From that moment, having never compensated in his life when overcompensating will do, Bruce vows to make up for years of calling his own son by the wrong name. He's so dedicated and sincere in making the effort to remember to call his middle son "Tim" that Jason just shrugs and resigns himself to also being Tim for the foreseeable future.
