Two Robins can work, but not more than two. You can have Dick Grayson and Jason Todd, but Dick Grayson has to be older. Jason Todd's death has to be what pushes Batman over the edge. IMO, Nightwing should be 20 when he's recruited as Robin in 2000. He's Robin for 3 or 4 years, and then has a falling out with Bruce, either over his personality or something. Batgirl's recruited within that time. Nightwing leaves after his falling out in 2003 or 2004, and Jason Todd becomes Robin. Jason Todd is killed by Joker in 2005, and Barbara Gordon is paralyzed in that same year. This sends Batman over the edge. A grief-stricken Bruce pushes away Barbara Gordon, in part because he doesn't want her to get hurt even more. Dick Grayson even tries to come back, but Bruce pushes him away as well, not wanting to bring them back into the life of crimefighting. Bruce doesn't deal with his grief and guilt well, and spirals down a dark path that leads to him killing criminals (as we've seen in BvS).
So, this is a Batman who actually ends up crossing that line, but it's not a line that he can't back recover from later. And to explain why he doesn't kill Joker, well, he immediately beats Joker to a pulp but still maintains his no-killing rule. So, Joker ends up in Arkham. But as the years pass, Bruce's guilt gets worse and worse and it's only after that point that he throws away his guilt. And I'd have that happen gradually. He becomes more and more brutal, until he ends up killing a criminal. And, instead that waking him up out of his grief and guilt, he rationalizes and justifies it. That'd illustrate just how broken he becomes, before BvS and before Superman shows him the light.
So, in that way, two Robins can certainly work. Bruce essentially gives the whole sidekick/ally thing another shot after Dick Grayson leaves, but it's when his two allies are hurt badly that he decides to give up on it all. It's pretty much a similar situation to how Bruce was depressed over Rachel's death in TDK trilogy, except this is a Batman who turns that anger against criminals and essentially lashes out in rage.
But this does it in a way where you have both Robins and a Batgirl while maintaining the proper timeline.