I can't comment on the TTC issues but wanted to tell you there is hope.
I too have BED. I could never lose weight in the medium term through dieting because even with being able to cut 1000kcals a day through dieting, only 1 binge a week would see me easily being able to consume more than 7000kcals in one session and I simply couldn't lose the weight.
The breakthrough came for me in learning how to cope with life without bingeing, this then allowed the dieting to work.
Many people confuse BED with overeating or assume there was choice in what I did. I was pre-programme to binge, it was done in a zombie like trance so once I'd entered into a session I wasn't conscious of what I was doing, let alone able to decide to try to interrupt it and limit the damage. BED bought me nothingness. I was the only time I could feel nothing and cope with life.
With the option of damage limitation being out of the question, the only way forward was to try to predict when a session would I cure and to interrupt the cycle to PREVENT an incidence. Again, people who don't have a clue had lots of well meaning advice about not keeping crap in the house and making better choices about what I ate. From what you've written, it sounds like you have more control and self awareness than I have. When I binge it's not a decision it's a trance. Even so, I think that what's worked for me might help you too.
I started writing a food and mood diary the moment I thought about or ate anything. To make it clear, it recording the intentions and thoughts, not just when I actually are anything. It would chart what time, where I was, who I was with, what I was thinking, what I was doing, what food was involved etc.
This allowed me to build up a picture of the pattern of my dependence on food. There was a very clear pattern! From that, I could predict and learn to avoid the situation and so interrupt the cycle. As time went on, the interruption of the cycle allowed me to develop alternative behaviors and see those as natural. I'd found a way to cope with life that didn't involve self destructive behaviours.
Last year, I only had a few times where I binged (it used to be at least a weekly occurrence). I do overeat as well at times but i have mostly learnt to manage my BED for enough of the time to remain a normal BMI. On the occasions I have reverted, I don't go into self hatred mode, I simply accept that I will always have BED and that each occurrence is my opportunity to learn and be better at coping in future. I am good enough at coping with my condition enough of the time now but I'll never be perfect, never be cured.
So for me at least, dieting wasn't my answer. It was only by tackling the BED that I could then diet to reverse the damage I'd done by bingeing. I hope my story might have elements in it to help you but from what you've written I know there are big differences between us.
Good luck