The members on this forum are predominantly students who will not have sufficient knowledge of or contacts within the legal industry to answer your question. I am a practising barrister but cannot ascertain from your post what type of claim you are looking to bring. I would be a little wary of assuming that direct access is the cheapest or best route to obtaining effective legal representation here. Even barristers who are accredited to undertake litigation (which is an additional element on top of being direct access qualified) do not have the same systems in place as solicitors when it comes to running and managing your case day to day. I would therefore reconsider whether or not a direct access barrister is your best option here, and I say that as a barrister who is direct access qualified. I do get requests on occasion for someone to instruct me on a direct access basis to effectively run litigation, and my response to those requests is to reject them and refer them to a solicitor (normally one that I know), because that is simply not a service that I can effectively provide, no matter how much you'd be paying me.
If you do want to continue to look into the direct access route, the best option would be to contact individual Chambers and ask the clerks to recommend any barristers that they have who are direct access qualified and practise in the area relevant to your claim. But as I say, if I were you I would be making enquiries of solicitors as well.