I think there are a few questions needed before anyone can really answer this:
You mention that you both raised issues of violence against each other but your report was NFAd; what happened to her report? Is there an ongoing investigation or have there been any charges? Are there any civil injunctions in place, or bail conditions etc?
Your brother being in prison; were you involved in any way in what he is in prison for? In what way are CAFCASS using this to prevent contact? What is their rationale?
I assume you’ve proposed changing your DD’s name to your ex and she has declined, but you’re asking court to make an order to force a change; what are your reasons for wanting to change? The court see the name of a child as part of their identity, so there would have to be able to see if there is a detriment to the name not being changed.
Why have CAFCASS recommended supervised contact after a period of letterbox contact? What is their rationale in their report?
CAFCASS can make recommendations like these, especially if there are outstanding questions around domestic abuse, but it’s up to the court whether to accept. If they have questions they will explore further and possibly go to Fact Finding; has this happened?
Essentially its in a process and there’s no real way to speed it up. Best thing to do is be compliant, show that with everything you do you are focussing on the best interests of your DD, and wait for the courts to do all the checks they need to.