Bella suddenly wants to know who this mystery person is. Maybe he has answers to her questions.

But it's obvious that Sherlock is deliberately shielding his identity, and wouldn't tell her if she asked.

"No," she agrees, "it's not. But if he is lying, and you can prove it, then you'll have a more accurate picture of the situation and you can decide where to go from there. You can't make an informed decision without information, no matter how much you don't like the information."