The Ex also isn't responsible alone for negative behaviour from dd, before he moved he had her half the week so his parenting will have influenced his dds behaviour just as much as his ex, you both seem to think it's totally ex's fault though so I'm wondering if the "bitchy" "bratty" behaviour has started after he moved away? The most obvious cause of his Daughters behaviour is the impact of her fathers actions and she's not old enough to have the language to express it. He's chosen a new sibling for his dd, a new second home (that she doesn't get to go to), and massively reduced contact with his dd so that he can have a new family. Did he really think that his dd wouldn't be disrupted by this. Why is the ex to blame? And to top it off he's missing the very little contact he has arranged because his partner couldn't travel, and because he didn't put money aside to see her.
If he doesn't phone/face time his dd regular, esp when he's missed contact then his ex isn't unreasonable to mention that. Also when he has cancelled visits what reason did he give his dd?
I'm not sure if he worked out travel expenses and moved away knowing he couldn't afford it, or if he just didn't consider how he was going to manage access before moving. Not sure which one is worse.
That love you feel for your ds, can you imagine moving away from him at four years old and seeing a lot less, can you imagine not going to see him because your partner is unwell, can you imagine not going to see him because you didn't have enough money, can you ever imagine not having time for him? Your partner is getting harsh comments because he's done all those things, he actions and choices are showing that his dd isn't the same to him as his other child. That she's not part of his life now that he has you and baby, and to an extent it's true, she's gone from being with him half the week, to moving away with you to have new baby and the expense of him not seeing her.
You say you think your dp is depressed, he's had big life changes too, could it be a combination of less sleep with new baby, new home, new area, and not seeing his dd causing his sadness rather than her behaviour on the very few times he does see her? It's unfair to make a four year old responsible for the feelings and moods of a grown man. If anything, the father is responsible for his dds negative moods and feelings, not the other way round.
It can be draining when the ex is awkward, and she may have done some horrid things in the past, but it sounds she's been left to handle the fallout of dp moving away, comforting her when she's upset at missing her daddy, and then if she's the one telling her little girl daddy isn't coming cos he can't afford it, daddy isn't coming cos and his partner can't come, as well as going from 50/50 parenting to picking up your dps share too then she probably is a bit angry and possibly stressed.
All the adults need to out this little girl first, through mediation if that's how it has to be. It may mean occasionally dp has to go see his dd without you and your baby because that's what's best for his dd at the moment.