I'm not arguing with the method of the TA, but with your idea that it somehow forms a compelling case for the truth of Christianity. Simply proving that Christianity is sufficient for logic does not prove that it is actually true--if you are going to argue for the latter proposition, there needs to be at least a reason why Christianity is probably true. In other words, you have to prove that the denial of Christianity equals the denial of logic necessarily.
If one is to do such an argument, would it not be better to prove that the denial of Christianity entails the denial of the law of non-contradiction, the reliability of sense perception, the law of the excluded middle, or the analogical use of language? The TA just isn't sufficient to prove that the denial of Christianity entails any of these.
Van Til accuses the classicist of assuming a fair standard of proof and indeed he does--because he assumes the ordinary standard of proof. If one were to use courtroom procedures to establish the resurrection, there would be no contest: the testimony would be sufficient to establish it. And that's the point: in denying Christianity, the unbeliever is assuming a set of criteria that would be absurd in any other context. The presuppositionalist wants to stack the deck the other way, while the classical apologist wants to even out the deck because he is confident enough in the evidence that is there and which the unbeliever denies.
I would agree that in principle, we don't even have to provide such evidence because we have no burden of proof. However, I would submit that in undertaking the task of apologetics, we are playing the prosecution, not the defendant and therefore we have the burden of proof. The battle is for the soul and we call upon the soul to "decide this day whom you will serve." In engaging in debate we call upon the unbeliever to judge: it is not God in the dock, but on the witness stand. And ultimately, the decider is whether the Holy Spirit moves in the unbeliever's heart.
My question about the TA is whether it would stand in a courtroom or even in a debate round? I don't think it would (at least not with me as a judge). At best it would yield a double loss where both sides have failed to satisfy their burden of proof.