I strongly disagree with people saying OP shouldn't say anything on the basis that adults need to keep the details of their relationship away from their children.
That's all well and good if both parents are abiding by it, but that's not what's happening here. The OP's ex is giving the daughter a provably skewed narrative and if the OP says nothing to contradict it, that's the narrative the daughter is going to believe, because everyone is acting as though that's reality.
It is MUCH healthier for her to learn that sometimes people tell different stories about the same set of events and it can be hard to figure out which story is closer to the truth. Over time she will learn what kind of evidence backs up a story (court rulings, behaviour over time) and this will aid her in developing critical thinking skills.
Of course the OP should be age-appropriate in what she tells her daughter, but she should still be clear that Daddy and Mummy have different views of events.
"It's not true that I tried to stop Daddy seeing you. Daddy was behaving in ways that weren't always safe, and a person called a judge - who helps grown-ups decide what to do in difficult situations - made the decision that Daddy needed to learn some new behaviours before he could see you again. He's done that, and I'm happy that the two of you can see each other again. But he shouldn't be telling you stories about the past that aren't true."
Personally I would also send a very restrained, calm but clear email to him (this is documentation in case you need it in future) expressing your concern that by saying things like "Daddy can't live without you", he is putting heavy concerns onto a child in a way that is unhealthy and may damage to her wellbeing. Express that she seems upset and feels the need to 'take care of him', which is not her role as a child. The parents are there to care for the children, not the other way around.
I'd say that you support them having a relationship now that he is behaving in more healthy ways, but do not support him saying things like "Mummy tried to stop Daddy from seeing you" rather than that the restricted contact was decided by an independent party. You have told her that what Daddy has said is not accurate, as it was a judge who decided Daddy needed to take some time away to work on some things first.
I'd also so that you believe it may be damaging for their future relationship if he makes false claims about the past, as your daughter will be able to understand they are false once she is older given that there is clear evidence to support what you have told her. You would be happy to leave the past in the past and allow them to build a good relationship, but BOTH parents need to commit to putting your child first by not discussing adult concerns with her from now on.
It’s like he’s hit reset and had a complete personality transplant.
There's already plenty of evidence in your thread that he hasn't actually changed at a fundamental level and is still engaging in manipulative and dishonest behaviours, but just so you know, this 'blank slate/full reset' thing is a very common part of the cycle abusers go through.
They tell themselves that they can simply wipe an unpleasant chapter or chapters in their lives away as though it never happened or was caused by something entirely outside themselves, and that they can start 'completely afresh' and 'be completely different' now. Of course, this 'history wipe' means they disregard their past experiences and therefore fail to see themselves when they start to repeat the same basic pattern, just with some new variations.