I can't see I've been through a situation quite like the one you mention. The first question that comes to mind is why would the HR department require that you claim the kids in order to get a benefit? They are still your biological kids and you are court ordered to pay for them. Are you sure your HR department has it right? It would seem a court order would trump the requirement to claim them as dependents.
As far as child support goes, I assume you receive a deduction because you don't claim them? While I think it's more common for parents to alternate the tax deduction annually, I've seen it done with a deduction plenty of times. However, there's quite a few things to consider when you don't get to claim them, not only the deduction, but the child tax credit, and the ability to claim head of household and receive all kinds of tax credits for living in poverty (as it would appear on paper that she does), and supporting children.