He was aware that he'd probably offered Veronica too much information already. He didn't want to expose himself that didn't need the information. He had done that with Lois, she was the only person he had actually admitted his secret to and that was only because he knew she would keep it despite her role as a journalist and they had come to an agreement. No more trashing the city and she wouldn't report the strange finding of the aftermath of the fight.
He had tried to keep his promise, though he had failed Lois twice since then.
When she spoke about the verbal altercation and keeping secrets Bruce nodded at her, "I can understand that. No matter how well you know someone though they do have the tendency to surprise you when you least expect it." Bruce knew all about keeping secrets. Dangerous secrets.
"At least he wants to protect you. It's noble, despite the circumstance and the outcome it may produce."