Ex is highly abusive but DD sees him on a limited basis. I have done my utmost to never criticise him and be neutral about him
I know this is the commonly accepted way of doing things but I still resent my mum for doing this. I hated seeing my dad. My mum kept encouraging us to go so that he wouldn't accuse her of alienating us. But it was torture (not physically but emotionally).
If he's an abusive man, why would you want to facilitate contact with your daughter? Why are you hoping she'll see his abusiveness for herself and load the responsibility on her to protect herself from the discomfort of seeing him?
It took me till I was nearly 30yo to break contact with my dad, at least 25y too many, frankly.
If I were you I'd tell my dd really seriously "do you enjoy seeing your dad? You don't have to if you don't want to."
I think it's deeply hypocritical when mums leave an abusive man, protecting themselves, but expect their children to keep seeing him.
Sorry if this is projection. But you've used the word "abusive". I'd do anything to protect my loved ones from abusive people. My mum did what she did out of cowardice and misplaced obligation.