Talk to him, explain how you feel.
"But DD doesn't help when she says things like "don't shout mummy" or "are you cross again mummy?"" She is smart, that is all, she can see you are cross. Sometimes it is OK to say "Yes, I am cross because I was expecting you to do X, Y, Z." BUt at three she is a bit young for all that so just use a very clear, non-shouty voice. I know it is bloody hard. She is too smart to be confused and cajoled by your shouting, you will need to use plain speaking and clear consensuses. For example the thing you are going to show her tomorrow could be her favorite DVD, and if she stay in bed she gets to watch it before lunch or whatever. It is not bribery it is incentivisation. I know it is boring to do all that and in an ideal world I would not bother but with kids sometimes we need to. How about a sticker chart, she stays in bed all week you take her swimming or to soft play while dad looks after baby?
"...it feels unfair because I live for that girl and I make sure we do loads of nice things together, she wants for nothing. And in the end I don't feel like she likes me very much sometimes." Of course she likes you, she loves you, she is just super smart! At 3 my dd said "Why don't you go and live somewhere else and I will live here with dad."
I was heart broken, I longed to hug her and say "love me!" But of course she did, she was just going through a phase, and I gave it no real attention. "I live here too." I said. And after a while she stopped saying it!
She is now 12, like you I live for my kid and adore my dd but she is bloody hard work. But she is sometimes rude etc because she knows she can be with me, and I will love her unconditionally.
I would say as much as possible stop the rude or not nice behavior with consequences BUT praise the good all the time, reward it, fill up on lots of fun times, and activities, lots of affirmation etc.
Think of it from her point of view. YOU and her dad are her world, but now a new squeeling little baby has arrived. Maybe she is feeling insecure. Make your dh step up and make time for your 3 year old just you and her.
XXX
She loves you, just as you love her. You are a great mum.