As far as the jurisprudence on this goes, in the cases Luther v. Borden (1849) (upholding Rhode Island using its royal charter rather than crafting a new constitution, even though the charter had strong restrictions on voting rights) and Pacific States Telephone and Telegraph Co. v. Oregon (1912) (upholding the use of referendums), SCOTUS has held the clause to be a non-justiciable political question. This was recently reiterated in Rucho v Common Cause (2019) (concerning gerrymandering).
So, in effect, anything Congress approves of when admitting a state is presumptively in satisfaction of the clause. Federalist No. 43 would seem in support of these conclusions, as it says "States may choose to substitute other republican forms, they have a right to do so, and to claim the federal guaranty for the latter." In this sense the clause would even be more a protection of the states against the federal government: they are free to choose any "republican form" of government, and the federal government cannot prevent this and must protect this choice.
Congress has, in fact, used this clause and the ruling in Luther to replace existing state governments, via the First Reconstruction Act after the Civil War. This law declared the governments of 10 states "unrepublican" and placed them under military rule. In Georgia v. Stanton (1868), SCOTUS again declared this a political question, effectively upholding the act (and other Reconstruction acts).
Colegrove v. Green (1946) further added that a state's malapportionment of state legislative districts cannot be challenged under this clause; it was later clarified that other clauses, such as the Equal Protection Clause of the 14th amendment, could be a valid vector for such a challenge.
