You seem to indicate that DSD's behaviour when she's at yours is 'ok', but that she 'reverts' at her mother's. Perhaps she's (rightly or wrongly) feeling 'pushed out' there and is fearful. Or perhaps her mother isn't being 'positive' about your baby. What is the relationship like between your DH and his ex? Because I think one of the best things would be if DSD's mother could be a 'partner' in reassuring DSD that the new baby isn't a threat.
I still don't quite understand what DSD's 'spiteful' behaviour is. Because if it's just saying the baby will wake her up/she won't sleep so she doesn't 'want it' that's probably partially based on her experience with her other sibling. Babies do cry in the night, it can wake the others in the house up. That's not 'spiteful', that's true and 'I don't want it' is an 8 year old's way of expressing it. Refusing to speak to her dad is her way of 'making him prove himself' by being upset or hurt at her refusal to speak to him. Again, not spiteful. A child's reaction.
All you can do is react with love, understanding, and patience. If she acts out physically, obviously that needs a 'penalty' along with reassurance. When she's with you, table any discussion about 'her and the baby' for later unless she brings it up. To say "you will be the best big sister", "let's do X together for the new baby" only reinforces to her that she is not the 'main focus'. There will be time enough for that later, closer to the baby's arrival. Let her feel what she's feeling for now and offer reassurance when he asks for it, either verbally or through her actions. You have time for that.
You will have a lifetime of similar experiences with your own child, trust me, so now is the time to learn. All parents have heard versions of "I hate you" or "You're the worst Mum" or "I'll never speak to you again" to which we have responded with "I love you" and "You're so special to me" and "I'm always here to talk when you feel like it". Of slammed doors and a tantrum or two to which we have responded with conversations, appropriate discipline and 'family rules'.