As a young woman, Marina would not have access to her dowry to make such arrangements. Her finances were completely controlled by her father. On marriage, her finances were completed controlled by her husband. And if she was moving among the ton in London, then she was also of that class at home in the country, and having a baby out of wedlock would have been just as much a scandal there. The only option for doing what you suggest would be for her to be sent away to a place where no one knew her to have the baby, which is what often happened in such cases. But it would cost a lot of money to set up such an establishment and a chaperone would have to be sent with her, so it was a fairly major undertaking. It is well established within the show that the Featheringtons did not have the money for any such thing - that's why Mrs Featherington approached the convent, but then it turned out they didn't have enough money for that, either.
Of course, what they should have done is inform Marina's father and let him make all the arrangements but for some reason this was out of the question - I think because of Mr Featherington's debt. Marina clearly felt that making her situation known to her father was out of the question - and if the Featheringtons were the lesser of two evils there, that probably tells us something.