Ironically, Babette's one of the most mature and well-grounded people in the group, right next to the Redguard dispatcher, thankfully, those are the two you get to balance out the jester.
Maybe 200 years ago she was all angsty about never having the chance to grow up, but she's past that now and comfortable with who she is and where she is.
That said, I only know this because my brother did that arc on his game. I always do the same thing: ignore the DB until they send an assassin after me, then track them down via the only available lead (the hit on Grelod), and then systematically wipe them out, one by one. The Brotherhood lives or dies by the contracts it accepts: you don't assassinate the son of the special forces leader, and if you want to assassinate a Nord named Basil, you better make darn sure you do it right the first time, because there won't be a second.