I very much doubt she was the one driving this although I don't know that of course. Poor decisions made a year ago doesn't mean she has to give up her child, though! That's not what she agreed to do and that's not what she wants to do, unsurprisingly.
Based on what? She says she was doing what she thought was best. Her poor decision to put a man who isn't the father on the BC is also a huge mistake.
Wether the child is blood related or not, is neither here not there. OP has been happy for him to be the father for 3 years, legally, and happy for the MIL to be the main carer for a year.
Legally, right now, he has PR. Even when the birth certificate is changed it won't be that simple.
If a mother removed a child to her parents and just allowed the father to see the child at weekends no one would be saying the man should go remove the child.
The op is not the child's main carer. She was a child when she had the baby or gave him to her MIL. Removing the child and undoing this will not be simple. He is the child's father, legally and to the child.
And I don't believe they could not sort childcare. Between 3 parents? Pull the other one. This suited the op and she was fine when it suited her.
She needs legal advice before she goes trying to remove the child from the house that is his main residence. Luckily, it sounds like the OP has more sense than most posters here