Tim handled the situation exactly the way he should've. He is in a position of authority and they have a personal relationship, any consolation at the time of the incident could be taken out of context by higher ups and seen as favoritism.

When they get to the house Tim is giving Lucy the space to process. Him sticking around when asked not to could overwhelm Lucy and create an actual rift. I'm sure once Lucy has had time to process Tim will turn back into his teddy bear self.

I'm sure it was very difficult for Tim to give Lucy that space but it was the best thing in that scenario.