Bloody hell, of course not!!
This is a very VERY good opportunity to set your stall out, and I cannot urge you strongly enough to take it.
You have a problem brewing here. Your ex's mother sounds absolutely vile - abusive and aggressive to you. It also sounds - unfortunately - as if she is keen to be involved with your child. Ordinarily that would be great. But it's not at all great if granny is a vicious, abusive witch who will interfere, demand, make your child a battleground and ultimately cause issues with putting you down to your child.
Where there is racism involved- I assume that you are a different race to your ex and therefore your child will be mixed race? - then the whole thing becomes ABSOLUTELY UNACCEPTABLE.
She's demanding to come to one of your medical appointments? Err, no. Don't bother playing nice. Hit hard and fast. Here's the kind of reply I'd be giving:
'Dear X,
As you might expect, the answer to whether your mother may come to my scan - a personal medical appointment - is no. It will be a shame if this means you can't be there, but if she is happy to blackmail you to the extent that you miss it at least it may give you some insight into her character which will be useful to you now that there will soon be a child to protect from her negative influence.
I find it difficult to believe that you would even think for a moment that I would be able to have her in the same room as me. To make this absolutely clear: this woman has verbally and racially abused me, has turned up at my house being aggressive and has had to be blocked from contacting me. I would not feel safe, let alone happy, having her at the scan.
It is clear from this that she expects contact with me and our child. This will need to be discussed between me, you and possibly a solicitor in detail before I will agree to anything. To reiterate: your mother has shown herself to be aggressive and dangerous, and also racist. I would be concerned for the safety of myself and our unborn (mixed race) baby if she were involved in our lives.
Please take this as a formal notification that I expect not to be harassed with any request for her to visit or be in any way involved with my child until a formal agreement has been reached, and it may be that it is the case that this would not be until our child is much older. In particular, I will state that I do not wish her to visit after the birth, and if she arrives either at my home or the hospital during the birth period I will call the police. I will be notifying midwives and the hospital of the history of harrassment, threats and violent language and will be requesting that she not be allowed in.
While I hope that we can maintain good relations for the sake of our child, I will not hesitate to also act if I consider that you are unable to keep our child safe from her. I would strongly suggest that you become independent from her as soon as possible, especially if her blackmailing behaviour is going to mean that you see less of the baby.
I hope that you can understand my concern and please be clear that any threatening or aggressive response to this email will be lodged with the police as the first stage in restricting contact with this person completely.
Yours...'