By "the South" I presume you mean the Confederate Congress. It's not clear what voters or citizens, North or South, thought of the suspension of habeas corpus. It's also not clear who could or who couldn't suspend the writ. The Constitution forbids the suspension of the writ "unless when in Cases of Rebellion or Invasion the public Safety may require it." As this is in Article I, the assumption is that this refers to Congress and suspension of the writ would be an act of Congress. But it's by no means certain. Lincoln acted when Congress was out of session. And an act of Congress in 1863 did provide for detention of those accused of disloyal activity to be detained until grand juries could decide whether or not to indict.
But surely the suspension of the writ itself, matters more than who did it. Congresses and committees can be as tyrannical as individuals, and there's never a shortage of representatives willing to vote greater powers to the executive with enabling acts. But I defer to those who have explored the question more deeply.
I don't feel the need to defend everything Lincoln might have done, but it does seem to me that the similarities between the war Presidents are greater than the differences.