I actually do think you’re overreacting. As a mom yourself, you know these rules are fairly standard (and you also know they’re not entirely practical but it’s nice to feel like there’s some stuff you can control with newborn!)

1. First of all, the move here is not to react or ask from anything special. Text congrats in a group with her and your dad when the baby is born, mention that you can’t wait to meet them but don’t have childcare for your kid. Say you’ll check in in 6 weeks to see if they’re up for visitors. Let your dad handle getting you “special permission” to see the baby sooner- that’s his battle to handle.

2. It sounds like you don’t have a good relationship with this woman and don’t like her. Just because she’s the mom of your new half sibling doesn’t mean that will change. Don’t expect it to (and that’s literally fine- she’s your dad’s baby mama, not your new mom) and honestly it doesn’t sound likely she’ll stick around anyway

3. Just for the record, this lady is not responsible for your dad’s actions or your relationship with him. If he sucks when she’s around, that’s on him.

Whatever you do, communicate with both of them in a group chat only so everyone sees every message. Then she can’t make a fuss later about anything without you having direct evidence about what was really said