Isn’t the requirement only that the government be “republican”? A republican government doesn’t necessarily have to be representative. It only needs to not be a monarchy.
That's the requirement of the Guarantee Clause (article 4, section 4) of the constitution- in its time, it was about barring non-democracy states from statehood, it was a guarantee of protection of any state from foreign invasion, and protection of any state from internal coup or rebellion.
But, if you look at section 2 of the 14th Amendment, it's a banger: if the right to vote is denied to citizens qualified to vote, the state doing it will lose its federal representation (as in, it will not just lose its electoral college votes in federal elections, its congressmen will not be seated). The purpose for this section of this amendment was to prevent confederate states from denying the formerly-enslaved the right to vote, and it should certainly apply today if Red-State legislators try to use their power to strip their citizens of their ability to meaningfully vote
For your consideration, here is the text of section 2 of the 14th amendment:
A literal reading of this text, apart from the anachronism by which voters must be male and 21 (which should be overridden by the 19th amendment, which enfranchises women's vote, and the fact that voting age today is 18) says that if your state doesn't let its citizens vote and abide by the result, its electoral college votes won't count either, and neither will its congressional delegation be seated.