How to manage hard to manage clients?

I recently have gotten a new client and I an their dedicated client manager. They are extremely hard to manage. They demand an insane amount of work and when told we don't have the capacity to complete it they say "just do it, we will qa it so you can skip the qa" and then we send it and they become upset that it is not perfect. When we tell them that was the agreement they tell us we need to increase staffing for their account. They already have 7 full time employees on their account. I've expressed to my higher ups multiple times we simply have no more capacity to do more work. Its understood throughout my organization that we are doing the very best we can, but we cannot put more people on this account.

I'm looking for tips and advice that help you when dealing with your overly demanding clients.

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