My ex SIL ended up in a relationship with someone like this, until my DB got a private investigator onto him and proved that he was married, just had a normal job etc. This was many, many years ago, well before OLD, and I think there have always been people who just get a kick out of having a secret, second life, sadly.
I have also known people who have had very hush hush jobs, and they have normal family lives, even if they have to travel and may be away for a few weeks at a time. However they have plenty of time off, go on holidays, live at home with their families etc.
As tumpymummy has said, once their DD starts school, they will want his contact details, for emergency purposes, and also because they will want to make sure that both parents, when not living together, are kept informed and updated. Likewise, as SummerGems mentioned, if his work is so secret and important, then he and his DD are potential targets, and the school will need to know and have a plan in place. My DD was at school with the DC of a couple of Russians, who were both significant targets due to their work, and the school had a detailed plan in place in case anything happened.
If she deep down knows that she is the OW, that she is effectively a single parent, then fair enough. It's her decision. However what does she/will she tell her DD, who will start asking questions at some point?
However if she really really doesn't realise that this is a complete web of lies, then it is scary. She is totally wasting her life waiting around for this man.
If they are engaged, then is the wedding being planned, have they set a date? I guess you don't want to push your friend away by keep questioning the whole set up, but you could keep asking her about the wedding, suggest she pushes to set a date etc.
In your shoes I have to say that I would get a PI on to him if I could afford it. Would you be able to contact her family to get them to help with the cost? I really don't think I could stand by and let this man ruin my friend's life. All you would need is for her to tell you when he was visiting, and then for a PI to follow him when he leaves, and build from there. In fact, would you be able to do that initially, or do you have friends who would be willing to do that?