PR, you have hit it exactly on the head, the cb is for the mother in her eyes.
She has 2 dc, dsd lived here with her father after her parents split (her choice) until she went to uni. She works FT, earns somewhere around 35K. CB is paid into the old joint account, dp already handles it, and the money is used for dss's gym, child care etc. It is literally a change on paper. I do understand, there are some ex's you'd think no way. But it's not like that here.
We have him wed- sat one week, wed-sun the other. Exactly 50-50 wrt nights. If you look at total hours then he's here slightly more, because I have him in the mornings to save paying for the morning club. CSA are not involved, it was an amicable-ish! split, before I came along, and there is no reason because it is so 50/50. The school question was first raised when she was facing redundancy, and I think she initially agreed because she did not qualify for any benefits, and I said I'd split whatever extra we got wrt tax credits 50/50. That still stands, natch.
I'm keen to switch it sooner than later if that is what needs to be done, because here the authorities are hot on checking for fraudulent applications, because we are in the grammar system. Going by the policy docs online cb is the measure they use. With the application we will have to submit council tax docs, and it's a bit dodgy if we're submitting docs from here and his registered address is there. That's if she gives dp a chance to make his wishes known.
the academy, from what I hear from parents with children who go there does ok, but has plenty of room for improvement. In addition a couple of years ago it introduced a new uniform which will require a mortgage to kit him out!!! The other 2 are good, but have different strengths, and better sports facilities. the one dss likes is heavily subscribed, and he stands no chance from his mum's address. a letter to the LEA supporting the application is an idea, but quite frankly, I'm staying well back, and leaving them to it. Dp is scared of being perceived to be acting in the way she does, and so is a little passive , but it is now his place to do anything if he feels it should be done.