4 is still so young, I have a 4yo DS and he is very much still a baby.
Honestly you sound a bit like my ExH... he had no children in his life and expected far too much of my then 4yo DD. He expected her to be perfectly behaved at all times but she was still so small and they can’t always regulate their emotions at that age.
I asked him so many times to leave any discipline to me and to not get involved but he tended to believe he knew best despite not yet having any children of his own.
To be blunt it got to the point where I grew to hate him and in the end I made him leave because I felt my DD was being bullied by him. However our relationship produced our son who is now 4 (we split when he was 6 months old).
Despite a rocky start ExH is now a wonderful dad to DS and regularly still apologise and talk about how he now sees he treated my DD. Now our DS is the same age as my DD was at the time we were when we split he understands how I felt and how wrong he was to discipline the way he did.
We are friends now and talk about it frankly.
Also I’m a SM to two girls who I love to bits but I find it hard so you do have my sympathy. I don’t do any discipline, I leave it to their dad. Our relationship is good, the kids love me and I find not being overly involved with “trying to shape them” or “fix” any behaviour I don’t like is the best way to be. Also I don’t expect my DP to get involved to that level with my kids either.
Take a step back and let dad deal with it. 4 is soooo young still, tantrums are normal at that age. She may be clever but she will still be emotionally immature and that can only develop with time.
Also she’s not manipulative. She is a very young child testing boundaries and ideas, it’s normal and certainly not indicative of a naughty child.