That was done, and rejected on "standing" and "ripeness". I understand the ripeness part, there is no injury for someone just being nominated by a party, eligible or not. Perhaps between the election and inauguration would be the "proper" time, although that was attempted as well.
The issue with the challenges that occurred prior to the election, and between the election and the inauguration, is that they had the wrong plaintiffs. I believe, based on this opinion, that the only real chance of getting a full hearing on a candidate’s qualifications would be for another candidate (major or minor) to bring a challenge prior to the election (or, as you mentioned, maybe in between the election and the inauguration).
Standing is a tough hurdle to clear, but it is an important check on judicial power.