I am putting my flame resistant suit on here but, having been in the DP's position, I can sympathise to a certain extent.
Yes, sulking and being a moody prick isn't on and I really hope your DD didn't pick up on it. If he's a sulky, moody prick in general then bin him off and no need to read any further.
BUT, when I first started a relationship with a man who had a DD a similar age to yours, he had 50:50 shared care with his ex, supposedly. But almost every single weekend without fail, the ex would back out of doing her share of the care - because she was going to a party or out with a new bloke or whatever. For the first three months of our relationship we only had two child free weekends together.
At first I tried to be understanding but after a few months of never having any time alone with my DP without the DD there, I have to be honest, my patience did wear a bit thin.
A new relationship needs time and nurturing to flourish. We could never go out because the child had anxiety and DP didn't want to leave her with a baby sitter. Even leaving her with his own mother, the child's grandmother was an emotional upheaval. It's hard when you meet someone you really like and want to spend time getting to know them on an adult level and there's never that opportunity.
I just don't think you can or should start a relationship with a childless person and expect them to be able to handle being thrown in at the deep end of a family dynamic where they're suddenly in loco parentis of a child every day. It's never going to go smoothly.
People say 'oh you know what you're getting in to' when you have a relationship with someone who's got a child. But in fact the opposite is true - especially if you don't have any children yourself. I had no idea what it was like to be around small children for longer than a few hours at a family barbecue. No clue. I found it really, really hard.
I'd pack my sexiest undies into my work bag on Friday morning along with a nice outfit for going out for dinner, maybe get a manicure on mu lunch break, all excited and ready to see my DP for the weekend only to get there and find his DD there. Takes the wind out of your sails a bit.
Obviously the child is always going to be the main priority but if you start a relationship, you should make them feel like they're at least A priority. Not that they're unreasonable and selfish because they aren't eternally grateful that you've invited them into a situation where your dickhead ex is now affecting their life and they can't say anything about it.
Anyway, hope that gives a bit of a different perspective.