If the man wants to be the father of the child I would suppose he has a right to it, and, barring a serious impediment, the woman is obligated to marry him.
He can be the father without being her husband! A women is never obligated to marry a man, especially not because she is pregnant. What nonsense. The only obligation you could argue would be that of a serious fiance who has pledged herself to a man. Even that "obligation" is based on old intricate social structures that looked on a broken engagement as shameful and put solemn betrothals in place to discourage them. Considering that almost 100% of wedding expenses are on the bride these days, it is the man who needs to be careful not to cancel things too late when the bride's family could be out thousands of dollars. Also, serious impediment...???? He is not a Catholic. That is already a huge, serious impediment. It should have kept this young lady from keeping company with him period. Unfortunately it did not.
I think the baby is being overlooked here. If the child is brought up without the presence of the father, and later finds out that the only reason he wasn't there was that his mother/family wouldn't tolerate his being non-catholic, that might make him want to leave the faith; which would be understandable, since the faith shouldn't overturn our natural obligations to our children, and as they've conceived a child the number one priority should be working towards an environment where the child can be brought up normally with its father and mother. When I speak of an obligation I'm thinking of the child.
I don't know why fear of future divorce should be an obstacle either. They've already had intercourse and conceived a child, so according to nature they're already married; separating now would be physically and psychologically speaking identical to a divorce. I suppose the fear of getting the civil & ecclesiastical stamp on the relationship comes from all the legal obligations that come with it, and that might complicate a separation/divorce should it occur; but then presumably the father is going to be financially responsible for the child no matter what happens. This makes me sympathetic to those who in this situation avoid any official marriage and just choose to live together and raise the child as if they were married – a kind of natural law marriage. If laws and social obligations (like big wedding expenses) are putting people off getting officially married, that's a sign these laws and social obligations are acting as more of a hindrance than a help.
The fact is as long as they have this child, they should do everything they can to give it a normal upbringing. What might happen in the future doesn't change that.
In Lithuania, it is the policy of the bishops (NO) that marriages cannot be performed under those circumstances, so it is a country-wide ban. This was apparently in response to a huge number of requests for annulment being based on a case that pregnancy had interfered with free consent.
Well this is the "problem", so to speak, with Catholic marriages: they're indissoluble. So yes, neither partner would want to put themselves in a position where if their spouse leaves them, they're forced to be single from then on if they don't want to be shunned by the Church*. So my advice to this couple would probably be to just carry on living together or maybe get a civil marriage, and if later on their relationship looks like it can meet the standards of a Catholic sacramental marriage, then they can marry in the Church. That seems like the common sense course of action. There's no point in getting a Catholic marriage if you don't truly intend to enter into that indissoluble, sacramental bond; that would be more or less hypocritical, and I sympathise with priests who are sick of people using Catholic churches to conduct weddings they don't appreciate or understand. They both need to have something of a religious conversion / catechesis before getting married in the Church. If this sounds scandalous, then maybe Pope Francis had a point after all in
Amoris Laetitia, about the disconnect between the Church's high and strict understanding of Christian marriage, and the prevailing culture which puts so few warnings and obstacles on entering into a carnal relationship. It's not just that people are only half- or quarter-catechised, it's that the culture itself is counter-catechetical. The Church's strict understanding of marriage developed in a strong Christian culture where people more or less knew their obligations, and where the cultural mores reinforced them.
* But then, at least among traditionalists, I suppose they're shunned anyway for having a child out of wedlock and living together. So it's a kind of damned-if-you-do / damned-if-you-don't situation: don't get married, and you're condemned as a fornicator, can't go to communion because you're committing a mortal sin every time you sleep with the person you're living and raising a child with; but if you do get married and later there's a separation, you risk being left alone for the rest of your life, or getting barred from communion if you do re-marry. So I suppose there's only two options: have a religious conversion and trust that God will pull your marriage together and make it work, and raise your child in the Church; or turn your nose up at the Church's legalism and have a lukewarm relationship with it for the rest of your life. Again, I think this is what Pope Francis was trying to work out in
Amoris Laetitia. For people in this situation, it's as though the sacraments are being used to push them away from the Church, rather than to draw them in. Yeah you can just well say, "they shouldn't have done it", but that's besides the point: the point is, how is the Church supposed to help them live a Christian life and bring up their child in a Christian house now that it has happened?