I think your problems with your dm (which are genuine) are clouding your views on your dd's birthday.
You're hurt at your mother's negativity and constant interfering, but also hurt that she won't be giving dd her full attention on her birthday. I think it might feel as though you're getting all the bad, and none of the good.
If your dm was not the way she is, I honestly think that you'd see the birthday thing for what it is; a newborn baby and mother needing support comes before an older dc's birthday (particularly the 1st birthday as dcs don't remember or care about them, they are for the parents only).
It is horribly to be bombarded by negativity. It hurts, and it takes the joy out of the moment ( because you start waiting for it all to go wrong).
I think you do need to sit your dm down and say these things to her.
"Dm, dd is amazing and brilliant and doing so well. I was worried about how I'd cope with it all, but I am, and I'm really proud of my little family. But there is an issue I really need to talk to you about. I love you, and appreciate your support, but I cannot cope with what seems to be constant negativity from you. If I want advice on something, I will ask for it. But when I don't, please don't tell me something is about to go wrong. I know i don't know everything. Let me find out. And if things go wrong, I'll come to you for help. But constantly feeling like I should be preparing for the worst isn't helping me. It's hurting me."
Then tell her again that you and dd love her.
Over a few weeks see how it goes. If she starts saying anything, think of a silly keyword to say, something like, "here comes the doom and gloom!" If she does start. Just cut her off, and repeat, "I asked you not to say things like that". You can do this in a jokey way. It's about breaking a habit for her.
If she really won't change then for your own sake reduce the amount of time you spend with her. You can tell her why and explain that when she starts being more positive you are happy to see her more.
I think in her misguided way she thinks she's helping. Make it clear what would help. Spell it out. Before it destroys your relationship with her forever.
And do be careful not put yourself in competition with your dsis. She sounds like she's sympathetic to you, and just ignoring your dm for now. If you could work together this would be easier. Don't begrudge your dsis having her dm around her when she gives birth or shortly after. We all need support at that time.