child benefit it paid per child, you can’t claim twice for the same child.
I know a couple who do shared care, they claim ChB for one child each and theoretically split the other costs 50/50, although in practice this is a bit of a push pull as one (my friend) thinks she is really hard done by and should be getting maintenance as she pays rent but the father pays a mortgage (no it doesn’t wash, she could have bought a property with the divorce proceeds like he did, she chose to rent and blow the money) and lumps all the costs on the kid’s father and thinks that’s fair. They both earn the same within a couple of hundred £ per annum.
I think the difficulty here is the idea that you’re used to the income, maintenance included, and that therefore you should retain it. Sadly that isn’t how it works, as your benefits are based on you having the children. If you relinquish ChB for one child to him, your entitlement to Tax Credits will reduce accordingly.
If he would be eligible for TC if he had one lot of ChB, I think there is a discussion to be had as to why you should keep claiming all that income when he has equal care of the children. If his income is over the TC threshold and therefore he would not benefit further than the £20/wk ChB, I think you should keep claiming as is and certainly should not give him any of the money.
If his income is higher than the TC threshold then you could still be due maintenance from him. My friend as above enquired as to the worthiness of making a claim for maintenance, and was told that although there was a small amount due, if her ex claimed the same would be due to him (because their income is virtually identical) so it would cancel each other out. Therefore. if your ex has a bigger income that you, there would likely be maintenance due to you, albeit a smaller amount.
Incidentally, I have just used the gov.uk CM calculator to calculate based on first my income then my ex’s income based on shared care, and there would be a small amount of maintenance due to me after both claims were netted off.