I don't think he'll get exposed next chapter quite yet, since he does in fact have a daughter once he says that she's much older than his wife is, the coworker might think that the picture is of the daughter when she's younger, and if she asks about it in confusion, he'll probably go along with it or say that it's his niece or something.
The setup for things going bad is the Golden Week date, as the coworker and the family are going to end up in about the same place (might be the guy mentioned it during one of their conversations or not), how that turns out could go a variety of ways (the wife trying to run away because she wants him to get with the coworker even though it hurts her, the coworker seeing the daughter and wife at the same time after being told the picture was of the daughter and figuring it out, the coworker and guy end up separated from the other two and end up on a mock date as they try to find the other two, etc.).