The Writ of Habeas Corpus doesn't come from the Constitution, it is from common law. However, the Constitution does say "The Privilege of the Writ of Habeas Corpus shall not be suspended, unless when in Cases of Rebellion or Invasion the public Safety may require it." This both protects Habeas Corpus, and also allows for suspending it. It was controversial when Lincoln did it because he did so unilaterally as president, rather than Congress doing it, but the Constitution doesn't actually say who can suspend it, or specify a procedure, so Lincoln's act was within the Constitution.
Tldr: Suspending Habeas Corpus is controversial, but not the same as suspending the Constitution.
This ignores the fact that the country is to the left of Democratic leadership, by far, on issues polling. Sure, if our choice was between right-wing neoliberal Democrats and fascist Republicans, it wouldn't be a contest. And in the general elections, that's usually about what we get. But these arguments are used, disingenuously, to convince primary voters to support right-wing Democrats, who actually do worse against fascist Republicans, by far. It's a losing strategy, and it's designed to be a losing strategy, because right-wing capitalist Democrats are more interested in suppressing their own party's left wing, than in fighting the country's radical right. They're more motivated by maintaining power than by preventing the collapse of American democracy. We have to vote for the best viable candidate in the general election, even if that candidate is a capitalist extremist Democrat, but we should be doing everything we can to remove them in the primaries, not just in spite of the general election stakes, but because of the general election stakes.
EDITED: Changed "issues voting" to "issues polling"