I love the fact that it seems Willy can't stand Babe at all. The contrast between him heavily flirting with him and how his demeanor changed so fast the moment Babe left the room… There is no more smile. He just seems so annoyed.

The way he clenches his jaw and how he seems to contain his anger. Everything in him suggests it is just a chore (or a mission). When Babe is around, he acts like the kind of men maybe he was used to hook up with before he met Charlie, but I bet it's not really who Willy is. He just pretends all the time because he knows it will give him what he wants.

Willy doesn't really try to flirt with Babe properly because his real target is Charlie. He wants to create a rift between Babe and Charlie and he acts the way he knows will work the best to trigger Charlie's insecurities. He seems to truly know what he has to do and it worked.

I mean look at his face when Charlie witnessed him molesting Babe (there is no other way to describe it). He knows he finally got what he was looking for and that's not hooking up with Babe. It's to show Charlie that he was right to feel insecure or to doubt. This will make it easy for him to accomplish whatever mission he has.