The state government is still republican under that scenario. That’s what’s guaranteed by Article 4. Even the President would still be chosen in republican manner, it’s just indirect. We elect representatives (our state legislatures). As our representatives those legislatures choose electors. Those electors choose a President.
You don’t have to agree with the actions of your representatives for a republican giver to exist. Personally I would vote against any legislator who wanted to use a method other than popular vote to choose electors. I’m sure a large majority would agree. That’s precisely why such an action, while insanely stupid, does not fall on Article 4 grounds. We the people do have the power to correct such an abuse at the ballot box and effect a change in the process.
The legislators cannot transfer their duties to citizens outside their state. To do so is to represent citizens other than those of their own state. It is quite obvious except to a lawyer I suppose who can infer any number of meanings to words written in plain English.
The same would apply to the so-called compact states regarding the electoral college. The electors would no longer be representing the people of their own state but the people of the 50 states as an aggregate based on the total popular vote.