Other mistake: When Alice asks Dan to make her some tea (after he tells her about him and Anna), he walks into the kitchen for just a moment, and comes back to find her gone. He also doesn't see her as he runs downstairs and looks outside. She couldn't have quietly gone down the stairs and out of the building in the short moment that Dan was in the kitchen. If she had moved quickly, there should have been a lot noise from her footsteps.
Other mistake: When Anna takes her first picture of Alice, she lifts the camera to her face, then the shot changes to faraway, and she lifts it to her face again.
Other mistake: At Anna's Exhibition, when Natalie Portman is smoking a cigarette, she's smoking a Red (Marlboro), but we can see the White Marlboro (Lights) box when she offers one to Larry (Clive Owen).
Answer: It appears so. Nothing else suggested he knew where she worked. More that he was just going to different clubs and drinking and came across the club Alice was working at.
Lummie ★
I think he knew exactly where to find her. He's a doctor and often doctors are known to have that "God" complex. I think he wanted to gain control of his relationship with Anna. To win Anna back he had to manipulate all the players including Alice/ Jane. He probably researched her and found her. That's why he knew Alice/Jane was not her real name. He was low key stalking all of them, to get Anna back.
At the strip club, Larry does *not* know that her name is legally Jane. He thinks her name is Alice. She is using a stage name at the club, and he wants to hear her say that her name is Alice. He refuses to accept "Jane" as the answer. He is waiting to hear "Alice."
But Jane actually was Alice's legal name. This is shown at the end of the movie, on her ID. Larry did not believe her when she told him.