Ah op, I'm sorry you're going through so much and at such a vulnerable time. Even when things are straightforward having a baby is so intense and the hormones are a lot to deal with so I think you're amazing for also navigating a breakup and all that goes with it on top of that.
I agree with pps that the first step is to sit down with your hv and be really brutally honest about how you're feeling. I had bad PPA and speaking to my hv was the best thing I could have done. She helped me see that I wasn't crazy, it was totally normal, but that I didn't need to just put up with it either. If your hv isn't someone you can go to then the gp is a good option too.
I think you need to think about who is in your support network and be honest with them how you're feeling as well. If they know you're really struggling they might be more forthcoming with support and even if it's not the dog maybe it's a bit of babysitting (even in the house with you there so you can sleep or shower etc) .
If money is really tight there are some animal charities who will support you to keep your dog with you, but i realise that doesn't necessarily resolve the issue of actually caring for the dog.
Can I ask what the situation with your ex is? Is he giving you money towards your child and is he safe/willing to take on some child care or maybe dog care?
Ultimately op, you know what you can cope with and what your limits are right now. If you need to rehome your dog to be able to survive for yourself and your baby then that really is OK and you don't need anyone else's validation to do the right thing for your family.
I would contact the breeder you got her from? I know we have an agreement with ours that if for any reason we couldn't keep our dog we would go to them first. Most rescue centres have the same clause if your dog is a rescue? Many animal rescues are inundated at the moment but you could ask that they put your dog on their waiting list or that you could keep her with you until they find a suitable home so she doesn't have to go through the centre itself?
When things are this hard op it really comes down to survival and rallying your support network whatever that looks like to you. It will get easier and honestly when you come out the other side of this and look back you'll feel invincible. You're doing amazing.