Sorry you've had a pile on, op. Definitely understand you are feeling awful about being stuck between a rock and a hard place. Everything you have said is believable based on your location and circumstances. Stop trying to justify yourself to judgemental middle class folk who have little clue of what life on the breadline entails.
My advice would be to accept the property but then submit a formal complaint following the councils complaints process.
Copy the complaint to your local Councillor and the Councillor responsible for housing. Councillors have far more influence over council officers than an MP who will be useless and possibly of a different political allegiance to the councillors?
Follow the process calmly. If you have any contacts at all who can help you navigate this system, use them. Be respectful and calm. Try to contain your anxiety- seek treatment and go to your GP as a preventative measure if you are feeling the pressure.
Consider taking out a restraining order against the ex. It is likely the only legal thing that would allow the council to act against its housing policy. If this isn't possible then I don't think there's anything you can do and if the council complaint process doesn't result in a better outcome you will just need to take steps to look after yourself and try to make the best of it. I know it isn't ideal to live in a place you don't feel safe but try to remember it may be your anxiety over thinking this.
Consider that what is best for your kids is a roof over their heads and take steps to improve your income, assess your outgoings, have a look on money-saving expert for good advice here.
Your anxiety will rub off on them so if you want your kids to be happy and healthy please try to focus on improving your own mental health by getting all the help you can. Sometimes it can feel like the world and the system is against you, and it is a bit. But think of 5 things every day to be grateful for and maybe take small steps to improve your circumstances for the future.
I wish you every luck. Document everything you do with official people, gp etc. Very important so you don't go mad when they run you in circles.