She is already pulling away.
This is a big thing.
It isn't just bedtime, it is a general thing, he is always criticising her.
The bar is always so low on MN, for so many.
In normal family life, children don't pull away from a loving parent that has to discipline them at times.
Children generally accept kind, firm boundaries.
This is not that.
He is an impatient man in a lot of his dealings with her, and the little 4 year old is now responding by withdrawing to protect herself.
She no longer feels safe with her father.
By all means defend this aggressive parenting if thats how you think children should be treated but I certainly wouldn't allow any man be constantly impatient, cross and critical of my children, allowing them to feel unsafe around one of their primary care givers.
The OP is arguing constantly with him about being too harsh.
Ffs, they only have one child.
He wouldn't behave like this with colleagues or anyone else, but his 4 year child is fair game.
I wouldn't be abandoning him to just get on with it, I wouldn't trust him from what the OP has written, to control himself and his irritation.
The child is not an experiment to hope he will calm down with.