Since the constitution leaves it up to the states to solve their own voting and vote reporting requirements, the obvious solution would seem to be an amendment specifying proof of citizenship before votes could be officially accepted by the Congress. This would usher in a huge debate about Real ID and big brother, but its a debate that needs to be had.
Will we allow non-citizens to not or not? The only way to guarantee they don’t, is proof of citizenship. As soon as this were to pass, the Dems would never win another close election, and it would doom their party. Naturally, you can see why they will fight this tooth and nail.