Gah, I didn't know the Tavistock specifically treated this but if you don't feel the Oxpip counsellor meets your needs (sometimes you have to try more than one person, as a lot of the benefit of therapy is dependent on trusting the therapist, I think) then they're world renowned. Definitely a fantastic option.
I didn't want to get into my own stuff earlier because I wanted to focus on the info, but for what it's worth, I actually only had that info because I didn't love my own son till he was 10 months. Between 2 - 8 months or so, there were times I genuinely hated him. I had hardly any of your challenges, either, just a ferociously difficult feeding relationship. He had serious problems with his tongue, that made breast feeding like having your nipple slammed in a door - a door fitted with razor blades. I was continually told I was almost there, and not to give up, until a specialist NHS clinic spotted that he had a problem that wouldn't allow him to feed without pain for me - ever. He was almost 4 months at the time, and my supply so established by pumping that I just felt obligated to continue until he was 7 months, so I hear you on that element at least, and how horribly an awful feeding relationship can affect your feelings for the little vampire at the boob. I once (completely deliberately ) let my breast flop over his nose while he was feeding because it was so excruciating, and I couldn't face prising his clamped jaws apart yet again against his fighting me to stay on (he was chewing the milk out rather than sucking, and nobody twigged until that specialist clinic). His terrified and heartbroken sobs as this life-sustaining boob suddenly tried to smother him will stay with me for the rest of my life. I don't think I have ever hated myself more. And lurking on the occasional thread here from people worrying that they didn't love their own child yet, being reassured by others who also took quite a long while to start, sustained me in hope, really. There will undoubtedly be people reading this feeling the same, although they never comment, if I'm any guide. Maybe someone needs to write that "Help me, I hate my baby!" book, hey.
I started researching the clinics I recommended because I was planning to use them myself (hence the child's ages being lower than yours), although Tess is a friend, so wouldn't have been appropriate for me. And then, as it transpired, as soon as I stopped feeding him the resentment receded and the love began to seed itself, and then grow. In many ways I was lucky - he was born a very chilled, calm, happy little boy and has remained that way, and I asked my Mum to move in and ensure that he never saw me when I was at my lowest, so I never really lost it before being able to hand him over to her so I could punch pillows/doors/walls out of eye-and-earshot. That meant that once I stopped feeding him it was easy to replace that anger and detachment with happy and positive interaction, and that in turn meant I was able to start loving him. You've had issue heaped on issue, none of which you had control over, and your son was and is just a more fractious child, too. Some just are that way and therefore harder to love, even when born in optimal circumstances - I have friends whose babies by any objective standards are very difficult ones, and it's really damn lucky they were born to mums who apparently worshipped them from birth. I think I would have found it really hard to build that bond with my timescale if my son had been like that, and I'd 100% have required outside help.
I suppose what I'm saying is that I would well have been you, if I'd had a tricker child, no mum moving in, and more initial setbacks than just the feeding horrors. I've only loved him properly 3 months, and already I can't imagine not loving him. And yet this love is so very glaringly a product of circumstance. When circumstances were horrible, and all I associated him with was extreme physical pain and failure, I couldn't bond. In a sense that has now killed my guilt over not bonding earlier, because, in retrospect, it seems so very obvious and logical as to why I didn't. In your circumstances I would absolutely and no question have been where you are now, only very likely without the courage to admit to it and find solutions as you are. If I hadn't known Tess I wouldn't have known where to start looking.
Obviously everyone is different, but I really hope that the range of women here talking about their own struggles to feel as they want to for their children will reinforce what your GP has told you - that you aren't alone, that what happened between you and your son is understandable and even in fact natural. And that it can be resolved, given the intelligent, determined and honest effort you are so plainly devoting to it.
You've both been so very unlucky, and you've had such a horrible time of it. I am so, so hoping that your courage and efforts are repaid tenfold and you can build the relationship with your son you so want and need to have.
Sorry this post is so long - now you know why I wanted to wait until you sounded a bit further into the process before swamping you like this!