Peachpie congratulations on your baby, and I am sorry about the circumstances around your step mother.
Does "I don't want her to have any involvement with our baby because of how she was (and how her attitude still is)..." literally mean she will have nothing to do with the baby or that she will not get a title like 'Grandma'?
I think in your shoes I would decide how far I wanted to take this and recognize that however you respond there may be issues with your dad.
So for example you can continue as you are or you could arrange to meet your dad, face to face, and A) explain, really explain, what your experience of your step mum was like and how this has influenced your feelings. - or not as you like. And
B) and/or Explain to your dad that you do not want his partner to be involved in your child's life. If you are not willing or able to explain why you feel this way it may be harder, but I think it is still do-able.
Decide what you will accept from your dad's partner and just say this is how it will be. If she is worse at home do not go to their home. Meet them at a neutral location or your place and be ready to exit the room or the house (with baby) if your step mum's behaviour is unacceptable.
I wonder if counselling may help you with the issues from your past, help you to process them and the affect they might have had on you?
I think you do need to be very clear with your dad. He failed you in the past and he now has a choice, he can support you, in relation to this, and he will need to tell his partner this is my daughter's choice. Or he can fail you again.
I think you could risk losing your dad over this but I think he could risk losing his daughter, his son in law (if you have a partner), and grandchild (and maybe any future children); so he really needs to be the one to put in the effort to make things work.
He can ring you when his partner is not around, or ring from another location etc. He can make it clear to his partner that he is visiting his daughter but she is not included, or he can visit in secret or at least with minimum fuss!
Your dad's partner will most likely make this an issue and want him to cut you out of his life. Which he may or may not do. Or he may choose not to discuss it with her and see you when he can. That's probably what I would do if I were him!
You are not asking him to choose between you and his dp (she may do that!). You are asking him to respect your choice not to include his partner in your or your baby's life.
What does your dp think?
Maybe in time you can find a way to meet with him, and she can be around but only if she learns to put others first (which her "full on tantrum and huff, tears etc lasted for days" would suggest is unlikely to say the least!
Just enjoy your baby, and protect him or her from toxicity!