Yes, the US government had the right to put down insurrection. This had happened previously, with Governor Lee of Virginia putting down insurrection in Pennsylvania, and Colonel Lee, the former’s son, putting down insurrection in Virginia.
Secession, the legal leaving of a state could occur by federal amendment certainly, or perhaps by legislation, or by federal court case with supreme court as original venue.
Successful rebellion could occur in theory, and be legitimized by treaty after the war. That is what happened in the American Revolution, but to do that, you have to win your rebellion. the Slave Power didn’t meet that standard.
Insurrection is not the same as secession. And nowhere in the Constitution is there a requirement for an amendment, or court proceeding for a state to withdraw from the Union.