Yes, it would be unconstitutional. But would it be a good idea? David Schleicher suggests that the answer is yes, and I’m inclined to agree. The basic problem is that voters base their decisions on down-ballot state and local races on their impressions of national parties. The result is that these races tend to be uncompetitive:
The implications of the mismatch problem are dramatic, as it leaves very little space for local accountability or representation. How state legislators perform in office may have a small effect on whether they get to stay in office, but, for the most part, their re-election chances will be determined by the popularity of the president. There is little reason to believe that officials who were elected because, say, the president of the same party conducted a successful war will accurately represent voter preferences on state issues. State legislatures are the workhorses of policy-making in this country, producing our contract and tort law, marriage policy, much of our criminal law, and lots more. But the content and effects of that policy don’t have much effect on state elections.
This would not matter so much if there were other effective ways to ensure officials were held accountable. But there aren’t.
I strongly recommend reading Schleicher’s piece, which helps explain the dismal quality of down-ballot public officials in my hometown. (We see something like Schleicher’s preferred system in Canada, incidentally.)