Larry: He's a stringy fucker.
Larry: I know who you are. I love you. I love everything about you that hurts.
Alice: So this man comes into the café today, and he says, "Hey waitress, what are you waiting for?"
Dan: Funny guy.
Alice: So I go, "I'm waiting for a man to come in here and fuck me sideways with a beautiful line like that."
Dan: So what did he ask for?
Alice: He asked for a cup of tea with two sugars.
Dan: I'm your stranger. Jump.
Dan: What's so great about the truth? Try lying for a change, it's the currency of the world.
Alice: No one will ever love you as much as I do. Why isn't love enough?
Larry: Of course she enjoyed it. As you know, she loves a guilty fuck.
Anna: Don't stop loving me. I can see it draining out of you. It's me, remember? It was a stupid thing to do and it meant nothing. If you love me enough, you'll forgive me.
Larry: But we're happy... Aren't we?
Alice: I don't want to lie. I can't tell the truth. So it's over.
Dan: I love you, and I need a piss.
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.