Huh? I thought it was a sign that he was way over his Dad's death to be able to take Bruce's name for himself like that.
I didn't take it as that. Though I honestly don't understand why with Tim writers seem to get into this whole "Wait, he's adopted by Bruce now...so he must be Tim Wayne!" when they've got the example of Dick Grayson never changing his name after he was adopted.
But I think in the Bat-family they generally just make it an ongoing thing that's actually more interesting for me. He's Tim Wayne because Damian makes him feel insecure about his place in the family, and he feels insecure about this sort of thing to begin with because of what happened to his parents. In Tim's case his dad issues were also always bound up in affirming his loyalty to one or the other and all that.
So yeah, I didn't see him over his dad at all, since most of his reaction to Bruce's death was guided by the way he's been reacting to deaths in the past. I would say the big step-forward moment on his dad for Tim, though, was in the Resurrection of Ra's storyline where he was tempted at the Lazarus Pit and Dick told him he had to let go and accept death. Now with Bruce he's found another way not to accept it (though he's also correct that Bruce isn't dead).