The constitutional way to suspend the habeas corpus is with congressional consent. Even Jeff Davis knew this; the Confederate congress twice authorized suspension of the h-c.
As did the U.S. Congress. But I will repeat that the Constitution is silent on exactly who may suspend it, and the Supreme Court has never ruled on legality of Lincoln's actions. Justices as recent as Rehnquist and O'Connor have admitted that just who may suspend it has never been definitively identified. So you can't just say that Lincoln was wrong.
And Jeff Davis is about the last person you should be holding up as a defender of a constitution.