When "meta" claims have implications within the system they are making assertions about, they collapse into that system. The claim that there are no objective moral claims is objective and has moral implications. Therefore it fails as a meta-claim and is rather part of the moral system.

The powerful want us to think that there are no objective moral claims because what that means, in practice, is do what thou wilt shall be the whole of the law. And, when two wills come into conflict, the stronger simply wins. This is why this self-contradictory position is pushed so hard in our culture.