Keep on it, don't give in - just be mindful that it might not go the way you want. Ask for a face to face with the GP, explain that you were tearful on the phone through frustration, not through self-pity at your situation.
Try to explain to her why you have previously binged, in my situation it was through emotions. Good and bad emotion would trigger me to go on a bender - then I'd feel shame and remorse, get over it and that would be that for months (until the next emotional trigger). I then did some CBT (Cognitive Behavioural Therapy) which was talking through my feelings and thoughts. It cost me about £600 but helped me think differently. I also went to AA, I'm not sure if I would ever be the kind of person to go through the 12 steps, but listening to others stories really helped my well being as I could see I wasn't alone, probably not a typical alcoholic in the general sense of the word, but definitely had an unhealthy relationship with the booze.
Something I have learned from my journey is that you NEED your GP on your side. Locking horns and complaining about them won't help, it will just reinforce their opinion.
I know your ex has swayed your opinion on what is and what isn't alcohol abuse and dependence, but try to step back. Think if a friend of yours was in your position, what would your advice to them be? Nobody here is judging you, reading all the comments I can see people have been up front and blunt (myself included - it's one of my traits I'm afraid!) but generally everyone is trying to be helpful and truthful, nobody has been abusive or trying to put you down.
You are quite clearly a very strong person, don't let this sway you. Even if you are refused until next year, just learn and move forwards stronger. Your story so far is pretty unique, most people don't get their licence back if the GP has issues - it is cruel that they gave your licence back then snatched it away, but it is what it is!
Take whatever time you can't drive as a blessing. 2 and 4 year old kids are amazing! When my kids were that old, they were the funniest people I have ever known! They said things as they saw it, they tried to stick forks in the dog's bum, they found the sound of ripping paper hilarious. Just put everything into perspective!