It sounds like you're trying to be very caring parents and you really want to get the balance right. My grandma always used to say, 'if they don't respect you when they're small, they won't respect you when they're big'. It's a very simplistic statement with a lot of flaws, but ultimately, parents need to set themselves as the leaders of the family when kids are small, and this can be expressed in as kind a way as you like, so long as you hold the boundary and don't get sucked into using pleading language or not following through.
I am an authoritative parent; somewhere in between gentle parenting and strict. I think most thoughtful parents slot in there somewhere. I fill my children with love, praise, connection, warmth and acceptance but I do expect them to follow my instructions without too much of a pushback.
If any of my children, aged 6, continually screamed and shouted at me, they would have a lot of their privileges removed and would get a serious talking to and reminded who they are speaking to. I wouldn't accept it, just as I wouldn't accept an adult speaking to me like that. I teach 6 year olds and honestly, they can handle having a toy/device/tv removed and they fully understand why.
If my child was screaming hurtful things at me, I'd calmly reassure them, tell them that you want them to stop using those words and explain that if it continues, they will get x consequence. If they continue doing shouting, remove what you threatened to do. If they behave themselves the following night, reward them and tell them that if they value x toy, they need to stay in bed and stop shouting to get her own way.
Every single time they behave like this at night, follow it up with a negative consequence. Every single time they do well, reward it the next day.
Don't over explain, don't get tucked into negotiations or 'please just go to sleep for mummy' type of weak language. Go in strong. 'I don't like the way you're speaking to me and I want you to stop now.' Don't cave in. Backlash always gets worse before it gets better when children get their first real taste of discipline after not having had much before hand.
Babies and toddlers I can understand, but at 6 your daughter does need a much firmer approach. I'm sure you will do a great job and you sound like a lovely mum with the best intentions.