Just so I understand this properly, is the reason you're speaking to him so often because he rings under the guise of talking to DD? when we all know he just wants to rub your nose in it
Does she want to speak to him on the phone? From my dim and distant memory of 6 year olds, they weren't 'good' with phone calls, so I wonder whether she's finding these a trial too?
What about an age appropriate way of asking DD how she'd prefer to have contact with daddy? I'm thinking that phone calls and having her round his house would be easy and comfortable for him, but 6 year-old favourite past-times like soft play or a wildlife park, probably less so. But if she's able to say that's what she wants- and what she doesn't want- there might be a win-win here for you and her.
If she doesn't want these phone calls for example, that means you don't have to speak to him at all, unless there's something urgent about DD or the divorce. You can insist on e mails.
FWIW, I often wonder whether there's a different slant on the stories you read about 'mothers refusing contact', 'using the children as a weapon' and other emotive phrases. Fact is, I can imagine a lot of kids don't want to spend time with a man who's never catered for their needs or spent much time with them when they were at home.
It's brave and honest of you to admit that you've been encouraging the relationship so as not to anger him into threats, but I'm glad you've realised that now and have had DD's needs ratified by a child professional and her own aunt.
The way to deal with threats like this is through the legal channels. I hope you've got a great lawyer who is used to little twerps like your husband throwing their weight around, issuing threats. He's a bully and like them all, will back down when someone bigger and stronger enters the arena.
Act smart here and get the lawyer on to it as a matter of urgency. If the one you've got has no appetite for it, find another one who does.
You also need to find a way of DD being in contact with him that doesn't involve you or your engagement. He's using these calls as a way to get your attention and it must stop.