Oh, wait, nevermind. I see you are getting that from the Full Faith and Credit clause. Well, I’m no lawyer, but I believe the second sentence in that clause is subsidiary to the first sentence, absent any other contrary indication, and only applies to “proving” the acts of the state to other states, so that they may know what they have to give full faith and credit to. You can’t extend that language to remove powers that are already inherent in the states independent of the Constitution, it’s ludicrous.
Article 4 says the Congress has the right to prescribe the manner in which a state proves its acts. Secession is an act of a state. Simple.