First of all, its unfortunate he wasn't 100% honest from the start, its tough to build a relationship on a lie.
I hate that I learned this third-hand. I feel like I am entrapping him - I mean, whether he is married or not isn't really my business. I am just worried that if he is being half-truthful about this, than he is being half-truthful about his girlfriend, I mean, wife's consent for his extramarital affairs. I dont want to hurt her or the child.
It is absolutely your business. If he is asking for a relationship (regardless of the context) you should know the truth. Heck I would even be so bold as to say it is well within your right to request some sort of confirmation from the wife, if that is how uncomfortable you feel. He has lied previously!
Personally, I am a ballsy person, I wouldnt have problem calling the guy on his omission. Maybe they have a DADT policy, maybe he has some reasons, but if you want this to go further than what it is, every card should be on the table
Honestly it sounds like you have the right idea, but there are some yellow flags that appear from the guy. (I won't say red flags, since you don't see them as a dealbreaker but they are worth noting to yourself) Protect yourself first.