3yr old Sleep-HELP

I have a 3 1/2 year old daughter. It's currently September, and it's been exactly 1 year since her sleep had been so out of wack. Last September I was 7 months pregnant so we were expecting some change in our daughter, because her life was about to change.

It's been a year since my daughter has slept through the night. SmEvery single night since she has been 1 I read her a book or 2 before bed and then i leave the room so she can go to sleep. Now that's she's out of a crib and in a normal size bed, she immediately runs to the door and acts as if she's going to knock the door down. Due to her tantrum and fits.
My husband doesn't budge, he has no tolerance for it. He says I need to sleep her and let her cry it out.
Sometimes after I read her a book I'll lay in bed with her as she falls asleep or I'll rub her back. She will fall asleep, but the second I gently get out of her bed she immediately wakes up, and then I can't leave. I usually end up sleeping in there or try again to sneak out until she falls asleep. I'm really trying to avoid sleeping with her. I have a 10 month old who currently still wakes up for a bottle in the middle of the night. Every little thing wakes her up, so I literally can't stay with her. Nor do I want to.
She also wakes up every hour of the night crying for me and if I don't go in there she will wake up the baby. So I immediately go in there.
My 10 month old sleeps better than my almost 4 year old.
I lay in bed crying because I feel so hopeless, and feeling like I'm a horrible mother because I'm the middle of the night I completely loose my shit on her. The lack of sleep I'm getting is breaking me.

Her daytime behavior has horrible! She's off her rocker! Part of me thinks it's because she doesn't get sleep.
She tells me "she hates me I'm the worst mommy ever" whenever I say no. She hates the word no. She goes in time out constantly in her room, and then will continue to try to bash the door down. Her behavior actually scares me and I don't think this is "normal" 3 year old behavior. She's more severe. Her tantrums are outta control.

I know a lot has changed within the past year for her. New brother and new room- a year ago. My husband travels every now and then for work. And she just started pre K full time. I talk to other parents and none of these kids wake up in the middle of the night. They all sleep great. And here I am crying for help and answers why my kid is up every hour of the night.
She sleeps with her lamp on, her sound system, so I know it's not the dark. It's literally severe separation anxiety. I've given her magnesium, I've given her melatonin, and NOTHING works.

Im extremely concerned about her sleep, and her behavior in general. I'm embarrassed and hopeless.

I'm thinking she may need a sleep study or something.
I'm planning on calling her doctor on Monday to try and get help or answers.

I am so desperate for sleep. And desperate for my kid to get good rest. It's causing my husband and I to argue, and I don't know how much more I can handle with the nighttime situation we have going on here.

Please, help.