I just don't know how to "play" into the emotions to keep it as stable as possible. The reality of it is, I do want what's best for all of us, including her. The emotions are what will cause damage in the long run and I'd like to mitigate some or all of it.
Have you spoke to an attorney about the process, including custody of your daughter? Believe it or not, they may have some practical advice regarding this topic.
Keeping the peace may mean that you treat your interactions in a business-like manner. Think of your 'job' right now as one where you are looking to make a deal with a business contact. You want the best outcome for you and to give enough that your deal is appealing to the other. Keeping the emotions out of it as much as possible is going to be a challenge.
What is the actual next step in filing the papers? What is holding up this process?