I don't think you should be pressured or feel as though you need to agree to this. It's an odd request and I've certainly never heard of such a thing? Especially one that comes from someone that isn't too fond of you. I don't think you're being unreasonable, you do what you feel comfortable with. If someone doesn't treat you right, don't expect them to have genuine care for your baby. That's a red flag. Personally, regardless of my relationship with someone, I wouldn't leave my baby with anyone. That's just me.
“We don’t need anyone to look after baby just yet, we will let you know if we do”. Done!
If you don't trust her, you don't trust her 🤷♀️ that said - there may be some good compromises here. Perhaps your husband could go spend some time with her with the baby. That way she isn't ALONE with the kid but is still helping out and she gets the brother time it seems you think she craves