Hi MeAndMyGirl. Got no idea TBH as I've never been in this situation, but this sounds as though it is such a momentous thing in your lives that I didn't want to read and run.
If your X has been denied access in the interim report then I'm guessing he's been abusive or has serious mental health or substance abuse issues? It's rare for the courts to deny all access.
I'm guessing that proof could take the form of police or doctor reports (if abuse), or even neighbours etc who may have witnessed it. If it's mental health or substance abuse, there will probably be medical records alluding to things? Unless I know on what grounds access has been denied it's a bit difficult to know, and I might be wrong anyway as I've no personal experience. Have you tried calling the Children's Legal Centre? They will be able to tell you exactly what it required in terms of proof and all the advice they give you will be free of charge. If you are worried about being cross examined, maybe Women's Aid can help?