So I did what was suggested and told my daughter that she had to see her dad this weekend and then told my ex that I was making her see him. The first day, they talked and she cried a lot, trying to get him to see how she felt and to apologize. She was pretty wrecked after a few hours and told him she had enough and he left. She was still pretty angry and felt that he didn't get it. He left things that she should call him and tell him whether to come the next day. So she asked me if she really had a choice and although I didn't want to force her to see him again, I did the thing that seemed right in the long run and called to ask him if he was really leaving it as her choice. He said yes and then I asked if it was going to be her choice as to whether she wanted to see him the next month, too and suggested that he might not want to set that precedent (especially after telling me that I had to force her to see him. I am still trying to figure out why that was my job and he didn't just let her know that he was coming no matter what). So he agreed with my point and said he would come the next day for a few hours. I know that all of this "flexibility" is in part because he really came down to see a woman he is interested in that lives here and not necessarily to see his daughter but if he visits our daughter, I have to pay for part of the trip. Nice, right?
So anyway, my daughter was pretty mad at me for not letting her take the out, but the next day they didn't continue to discuss their issues, just sat around playing cards and watching the olympics for a few hours. And afterwards my daughter said that something I had told her had really helped; that it wasn't her job to teach him a lesson. She said that really helped her feel like she didn't need to keep going over stuff until he got it. That she had said everything she wanted to say and now it was up to him to understand it.
Based on what she told me of their conversation, he really didn't get it. He asked for my input before hand and I told him that it would go a long way to cop to that he could have handled things better, but he never seems to want to see his part. Even when his 14 year old daughter is telling him that she is afraid of him, he started by denying that he never yelled at her. (Hey, everyone yells, but he actually rages and it IS quite scary) He did finally admit to her that he needs to manage his anger better, so I guess that is progress.
So thanks to everyone who helped me get to this place. He still told her that there needs to be a modification to the custody arrangement, but at least I did my part to uphold what we currently have.