It won't last forever. And I think you know that, deep down.
You sound very sensible, just at the end of your tether, which is completely understandable. She's tired, you're tired. You poor things, it must be horrible for both of you.
So it sounds as though if she doesn't have enough sleep, she plays up. Probably because she is over-tired. You mention that your DH wakes early for work - if this is disturbing her, can he be a bit quieter? Use a different bathroom (if you have more than one) to wash etc.
Can you incentivise her to stay in bed until 7am? I give my DD1 (4yo like yours) stickers if we need to re-establish good sleeping. She gets to put them on her bed, so it's a permanent reminder. I show her them the night before, so she knows what she's aiming for. Or if stickers won't do it for your DD, how about a bag of little pressies (much as you may have done when toilet training - I know I did!). If you go this route, let her see what goes into the bag - little books, hair clips, playdough - then she gets one if she stays in bed until 7am.
You don't say what time she goes to bed, but could you move it earlier on the days when she gets up early? Or persuade her to have a nap? I'm guessing by her age that she must be at school, but at the weekends, she could play catch-up. Even half an hour quiet time in her room, lying in bed, maybe with some books will calm her down and break the cycle.
Otherwise can you change her routine? My DD1 did this kind of thing last summer (possibly triggered by the arrival of her sister a few months previously - is there something similar going on for you, I wonder?). I was, like you, at the end of my tether, crying about it, wondering how I would cope and where my lovely DD had gone. It sorted itself out when she started pre-school in September. She needed the change.
If it is to do with latent jealousy over the arrival of your DS, can you have some special time with just her? When your DH gets home, take her out on her bike, or do some jigsaws. She may be feeling that she used to have all of you, and now she doesn't. It's a difficult thing for a four year old to deal with (and verbalise, hence the bad behaviour).
I hope some of this helps you. But it will pass. Honestly. And you sound like you're doing a brilliant job in very trying circumstances.