Hi Hun, I am at the same point to get things for my GRC as you, except I also have my GRS next week so I decided to focus mostly on that for now. I have the part of the application that my consultant was meant to fill out ready and filed. I still don't have some of the stuff that has to be signed by a lawyer. There's no need to overwhelm them with proof of your new name or so I've heard. They like to see proof of the name change itself and only 2 or 3 supporting documents preferably that are occupation or bank linked. So bank statement. Payslips from your work or college certificates things like that. They will want that letter you have from Bellringer. See if you have had GRS, that's when they only need one specialist's report.
I'm afraid I can't help a whole lot being as I don't have mine yet either. But we can always bounce ideas back and forth if it helps. I'm sure someone else will be able to let us know from their own experience.