I do know a few women who have separated from their partners while pregnant. In all cases, it has been impossible to discuss custody and keep the child's best interests at the fore until the baby arrived.
I know one women who formula fed and therefore allowed overnight contact early on after pressure from ex-partners family. Her baby had severe reflux. He ended up in hospital and under cross examination from an A&E doctor trying to figure out what the F was wrong with the poor baby the partner admitted he hadn't been giving medication correctly and hadn't been following the advice for feeding. Basically thought the mum was being a drama queen
Poor baby was in agony.
I also know a lady who again encouraged overnight contact early on as she felt it was the right thing to do who's baby reacted to the formula the Dad had decided to buy and it took a long time for the mum to work out why the baby was in such bad sorts every time it came back from dads.
My impression - sorry OP - is that dads who push for lots of alone time early on seem to be so keen to trial their way of parenting that they don't listen to the mums. Which can cause problems or be unsettling for baby.
The situations I've seen where it's been amicable are those where the Dad was super respectful of the 4th trimester and early mother baby bond.
Including one instance where the Dad ended up as primary carer once child was 2 as for various reasons it was clearly in the very interests of the child. Doubt that would have come about if he had taken an antagonistic stance about his 'rights'.
As someone else said - right now it's not about his rights, it's not about him. It's about prioritising the baby and thinking about the responsibilities he has towards the child. His first and most important responsibility is to support a secure attachment to the mother as that is clearly in the best interests of a newborn.