I'm not in this position myself, but a family member is. Carers Allowance should go only to the claimant. There's no question of splitting that. Even if you both meet the requirements and the other parent can't claim Carers because you are so is on eg JSA, they're still receiving some income.
There might be an argument for apportioning some part of the DLA to the other parent depending on childcare split and when disability related expenses will be incurred. So if there was a particular provision you wanted your child to attend and it was on ex's time, it might be justifiable for the DLA to pay for some/all of this. Possibly.