Hello again Thirtyeight
When I escaped DV I went to Relate first as I wanted to understand why I, as an intelligent, confident, well educated person with a responsible and difficult job (I work with teens who have problems of various kinds) managed to get myself into an abusive relationship. They put me in touch with Women's Aid & also helped me to understand that it wasn't my fault and that I had no need to be embarrassed or apologise for my situation. I was the victim, it was my ex who had the problem. I also realised that his mother not only knew about it, but enabled the abuse by turning a 'blind eye' & asking me to make allowances for him as he'd been spoiled

The first solicitor who I saw (before I contacted Womens Aid) was unhelpful & didn't grasp the problem. He said that I had no choice but to allow my DS (then an 8 week old bf baby) to stay every other weekend with my ex. He said that the courts would insist on this & suggested, while bfing, I could stay with ex during the contact weekends 
You really REALLY need to get in touch with a professional organisation who can support and advise you. For one thing, your ex doesn't have to transfer the Benefits over to you. An expert can support you to explain to DWP/HMRC that you are fleeing DV and that your ex isn't co-operating. Believe me, DWP deal with this kind of case daily, as long as you can show that you have the children with you, they will stop paying the Benefits to him & transfer the claim to you.
Alternatively, you could report him for claiming fraudulently if you have the children with you & he retains the Benefits. 
My ex literally hounded me for years & used the courts to continue his abuse. But, with the suppport of Women's Aid, some good friends, a counsellor & an excellent solicitor I won.
As result I'm a stronger more confident person &, possibly, a better mother
I'm certainly better at my job!
It's time to be a tiger mother & fight for your kids - GO GIRL