Would also like to add that he's three quarters on the way to ruining the relationship between you and your parents. I imagine it's been hard for you and them, ignoring the elephant in the room that is the disgusting behaviour of your DP.
Your parents sound great, they've obviously done a lot for you. Are you really going to choose him over your mum and dad? He's going to destroy your relationship with them first. Then it'll be only a matter of years before your DD is completely alienated. I'd give it ten years,max, for the damage to be done irrevocably. Just enough time for your DD to hit her teens and work out she can't rely on you, for your parents to grow too old and too distant for the relationship to be repaired, maybe for him to get you pregnant so you are extra stuck.
Sweetie, you've made a classic mistake. You've come out of a long, abusive relationship and bang into another one.
What made you think a bare year and a half was enough time to get over a decade's head fuckery? It's not your fault - you won't be the first or last, easy mistake to make, we all want the happy ending.
What will be your fault is if you make the wrong decision now, and walk open eyed into a situation you've been warned about. With a man who is actively nasty to your DD.
Take a step back from the follow your heart shit that women are spoon fed from birth. It's designed to keep us in our place, in situations exactly like this. Ignore your squidgy feelings and your hope that he might change and look at this coldly,with logic. What would social services/a teacher/a sensible friend make of this situation if they came across it?
Actually, that's another thing. Given that you've involved safeguarding previously, how do you think it will look if you marry this guy, have a baby with him (and his behaviour will absolutely escalate then- think back to your ex, I'll eat my hat if he wasn't worse during pregnancy) and imagine what will happen if DD makes a disclosure in school etc. And they pull up your file with its past history of abuse. They'll think you're unable to protect your child and put her needs first.
There are thousands of bloody lovely men out there who won't behave like this.
Even if you never meet one, you'll be far happier as an independent, financially secure, autonomous single woman, with a great relationship with your DD and perhaps a fuck buddy on the side.
Jesus, don't waste your life on this twat