We sure haven't seen her react at all to Bruce's death and hasn't she even blown off Dick for wanting to talk to her about it?
In the Oracle mini-series, Dick's message did touch on his being upset about Bruce, but it seemed to be mostly because Babs was being a hermit and not talking to anyone so he was worried about her. (Which, is pretty jerky to Dick, even if she wasn't also grieving. It's understandable to sometimes forget to stay in touch with friends, but when the friend's father fucking dies, you get the hell in touch, especially if you knew him and were friends with him.)
She's a very private person and so is Bruce. She's not welcome to do that any more than he'd be welcome to come into the Clocktower and make it his and his new person's base. He wouldn't. He's too independent.
This is true, although she has sometimes worked in the Batcave when working together with the bats on a case, I believe. I do feel like the batcave, while being Bruce's home office at its roots, has also functioned a little like the family room or something. In that, lots of people have met and worked there.
So there is some argument for it, but I do think Babs likes her creature comforts more than Bruce. She'd want to be working out of her home where she can just roll to the kitchen or go have a power nap or something. Bruce at least had his home upstairs and had Alfred bringing him food and whatnot. With Babs now, she's just sitting in a cave. Not as comfortable.