I'm just wondering where you got to with your claim?
I am going through a similar process (Rival CB claim) and have found a very useful High Court case Ford vs HMRC. The outcome was that where 2 separated parents provide day-to-day care to 2 children, then each parent should receive CB for one child. The number of days/nights does not need to be equal. The children just need to be living with the claimant in "proper way" - ie. have cloths, bedroom, eat, sleep, holidays etc. - so not just contact.
I am currently at the stage of providing evidence to HMRC - this includes a form they've provided, and letters to confirm that my address is recorded with the schools, GP, dentist etc.
I'm happy to let you know how I get on.