JB is where he is because currently the system sees him as a mass murdering child killer. I think his conviction will be quashed. Whilst his conviction is upheld I don't see his category changing unless his health/mobility requires a different environment and then he will no doubt be moved on health and safety grounds.
Prison categorisation is about the public risk associated with escape, and the risk presented by the offender also comes into it. Since Bamber is under a whole life order, that will inhibit his progress somewhat, but I suspect any attempt to block his progress on that basis alone would be considered unlawful, even if codified into the relevant Prison Service Instructions.
I don't think any of us know exactly why he is still in Category A, but the point is that Bamber was previously in Category B and this was changed back to Category A, apparently after lobbying from relatives. I personally would be making a big issue of that. I see no reason why he can't be in Category B and even lower than that, given that assessments of him confirm he is low risk. We needn't assume he is guilty or innocent, it's a separate issue. I have no idea what his solicitors are doing about it, but as far as I can tell nothing is mentioned on his campaign website about any campaign or legal moves to have his categorisation lowered so that he can make progress. And I wouldn't necessarily assume either willingness or competence on the part of solicitors to raise such issues. That said, he does (or did) have a specialist prison lawyer acting for him.
Despite him being on a whole life order, I think a re-categorisation is possible: Myra Hindley, for instance, was held in a Category C prison. The Open Estate was introduced during the post-War period to house and accommodate prisoners like Bamber, who have served their time and need to be re-integrated back into the community.