Amar and Chafetz basically say that the definition of 'returns' in Art. I, Sec. 5 of the Constitution gives the Senate the absolute right to refuse any return certification that they wish with a simple majority. That means that the Senate could reject Burris's appointment and certification (assuming that Jesse White is forced to certify the appointment, which he has heretofore refused to do).
McCormack didn't cover anything like this. In McCormack, the House of Representatives refused to re-seat a legally elected incumbent because they thought he was a scumbag. The Court determined that no matter what Congress feels about a particular candidate, it is barred from refusing membership simply because it doesn't like someone. The houses are limited, when judging qualifications, to determining whether an elected member meets the constitutionally mandated requirements. The case doesn't even consider "elections" or "returns", the other two items that each house has the power to oversee.
When weighing the issues present here, I honestly don't know what the Court would do. The Constitution does not lay out a burden of proof for the Senate to meet before refusing a 'return' other than a majority. By that, I doubt the Court would create an arbitrary burden. They don't like involving themselves in the 'political branches' anyway; it's unlikely they'd begin setting random parameters like 'reasonableness'.
However, the Court may consider the case in terms of due process. The due process of the law was followed here: the governor retains his powers and lawfully appointed a replacement. By refusing him, the Senate would be abrogating due process. The limit then placed on the Senate would be that they could refuse anyone who doesn't meet the constitutional qualifications or who wasn't appointed/elected through the due process of the law. Basically, the argument would be that the Senate has the right to refuse an appointment, but even that power must be constrained by the principle of due process. What's the point of elections if the Senate can refuse the results for any bogus reason? However, the rebuttal to that argument is that the Senate's ability to refuse and appointment is simply another step in the process.
I don't know how much water that would hold, but it's an idea.
Ultimately, I agree with Concurring Opinions, which says that [bold mine]:
My best guess is that the Senate Democratic leadership would argue that the Senate's authority to judge the elections of its members extends by analogy to judging the appointments of its members; and that a corrupt election would be cause to not seat someone, so a corrupt appointment should be too. But surely this sort of determination would require some sort of investigation rather than a conclusion that Burris is unfit for office (even if the Senate could get away with this constitutionally, it shouldn't try to). Burris has not been connected to the corruption case as far as I know. What are the odds that Blagojevich would appoint him corruptly in the middle of this investigation?
If I were a leader in the Senate, I would confer with Sen. Durbin and Illinois state officials, and see what they think. I might hold some hearings to find out more about the circumstances of Burris's selection. But I would not say that the Senate can just refuse to seat Burris.