Not rtft but HAVE read OP's posts.
What occurs to me (as someone who didn't have the easiest path to motherhood themselves) that it's possible she may have experienced pregnancy loss, medical issues during the pregnancy that made her at risk of losing the baby, a difficult/traumatic birth where possibly there was a risk of losing the baby...
Could also be that unbeknown to you the baby has been poorly since birth and that's why she is feeling so protective, doesn't need to be anything major to trigger that instinct,
Not everyone tells even those closest to them (sometimes out of a sense of protecting THEM) if they have had these issues.
I told nobody about my 1st miscarriage until my 2nd pregnancy almost a decade later, not everyone was told about the 2nd mc, during 3rd pregnancy (dd) I didn't tell people everything of what happened with that, as it would have just worried them and there was nothing they could do to help anyway.
Dd is now 18 and a recent conversation with my mum about a cousins current pregnancy difficulties (cousin has opted to be more open - entirely her choice) meant we realised I hadn't told mum until now of a couple of similar issues I had during dds pregnancy, I'd forgotten I'd not told her.
So bearing that in mind and absolutely not saying you shouldn't care for and keep an eye on your sister for potential clues to pnd or other mh issues, for now be respectful of her wishes but be supportive of her?
Tricky situation that needs careful handling.
However, all that said, I will also be honest that becoming a mother triggered a flare up in ocd for me which meant I was very protective of dd, but it wasn't just the ocd, having lost 3 before her, almost losing her during pregnancy and childbirth I think it understandable to have a certain level of anxiety and protectiveness.
My ex - who definitely didn't have any anxiety mh issues - but who was the father of pregnancy 2 and dd and was the one supporting me throughout the pregnancies, attending scans and all the other tests, seeing how sick I was at times, visiting me in hospital, almost losing me to the 2nd mc, visiting me in hospital again before and after having dd when I was on complete bed rest to avoid going into early labour, then a difficult & risky labour and what he himself described as a "terrifying" experience of almost losing us both...
He too was quite protective of dd and not completely comfortable when someone other than one of us was holding her.
4 months is still very early days.