I haven’t read through all posts but just wanted to point out a few things to think about. I know you say that your wife doesn’t know how to track down this guy, but I think she should make a good effort in trying. Go to the bar and ask around. This guy might be a regular and somebody may know him.
First of all, the guy has a moral right to know that he may have a child. Second of all, it is in the child’s best interest that he is found. Especially if this child is of mixed race, it could cause considerable emotional distress on this child to grow up not knowing where he/she came from. Biology matters. To some more than others, think about the possibility that this child could grow up being one of those people that it’s really important to? Would you rather tell her “this is the guys name” or would you rather tell her “we have no clue who he is, you were the result of a one night stand in a bar?”. Think about it. I think that the latter will be very painful for her to hear. This child has a right to know who her father is. So make an effort to find him for her sake.
I also think you should consult an attorney to protect yourself. This may not be the first time and it may not be the last that your wife strays. I know that’s hard to hear. You haven’t had time to truly heal from her adultery and I suspect that when this child is born, it can cause a lot of strain on your marriage and she may divorce you. You don’t want to be on the hook for a child that’s not yours for the next 18+ years (including college expenses). And I also think that you shouldn’t be on record as the legal father if you can’t fully 100% love this child as your own. It’s not fair to the child. Believe me, the child will grow up suspecting something is off with you and it will cause her pain. So you really, really need to get this figured out. Get some counseling, talk to your priest. But do not attempt to raise this kid as your own unless you really can put everything aside and love this child unconditionally as your own.
I know marriages can heal from adultery. It’s possible. But the healing process takes TIME, and unfortunately your marriage doesn’t have that luxury right now with this baby on the way. So be careful and consult with an attorney.