She could be trying to avoid telling the probably quite combative person on the phone that the reason you don't change the contact details on the MIS on the basis of the other separated parent's say-so is that could be a safeguarding issue - as I think it's quite clear that your reaction to that would be quite belligerent and that in turn could put the other parent and possibly the child at risk. And her job isn't to sit there and be berated for following Safeguarding procedures - after all, her life would be far, far easier if she could have said 'oh yes, I'll do it now (tap, tap, tappety-tap, click, click, click, save) - all done', rather than have to deal with you.
Have you had any contact with the DSL at all? If the DSL, knowing full details of the situation your child is living in, is happy for you to be added as an equal 1st contact, they can authorise the support staff to make the change. It's not a legal requirement, but some schools adopt a specific Separated Parents Policy - if you were to contact them school and ask politely, you should be able to receive a copy if it isn't available on the school website itself (again, it's not a legal requirement to do so, but some schools do it so that any caller can see what their policy is). Others deal with acrimonious divorces and issues on a case by case basis.
If you were to obtain an order stating where the child is to live, that would be of assistance (your ex would still have Parental Responsibility and the right to be equal #1/have all correspondence and messages/right to attend appointments/consent to things, but your child's address would be changed to yours) - the decision where to pay Child Benefit is not dependent upon that, though, as they can decide without needing a Court Order on residence.
If, however, Child Benefit do not feel that it should be paid to you despite their lower standard for decisionmaking and you have not ingratiated yourself with any member of staff who has encountered you, then your chances of enforcing this through sheer belligerence and some rather unpleasant attitudes that I suspect are made very clear whenever you call, are minimal.
Essentially, don't be a dick to people who a) are experienced and do a lot more all day than sit about waiting for your call to give them something to do, b) have your child's interests and safety as their priority, c) would far rather not have to deal with your attitude in the first place and d) if you were to give them what they need, rather than have a go at them, would be perfectly happy to give you equal contact ranking.