The idea of a "drone court" would send federal courts into areas they have never gone before, and indeed from which, I think, the text of the Constitution bars them. It could also put the integrity of our court system at risk.
Let's frame the issue properly. The present administration does not claim that the president has "inherent authority" to attack anyone anywhere. Instead, from the documents and speeches we've seen, the administration says it can order drone attacks only as provided by the Authorization for the Use of Military Force passed by Congress after the September 11 attacks--that is, against "those nations, organizations, or persons he determines planned, authorized, committed, or aided the terrorist attacks that occurred on September 11, 2001, or harbored such organizations or persons, in order to prevent any future acts of international terrorism against the United States by such nations, organizations or persons."
Unlike the fictional President Bennett in Tom Clancy's Clear and Present Danger, then, President Obama can't suddenly send the drone fleet down to take out, say, Colombian drug lords or the Lord's Resistance Army in Uganda. That's a marked change from the overall position of the last administration, and it's an important limitation on the president's claimed authority.
But because of that limitation, a court would be supervising the president's command decisions in a time of authorized military action--after, that is, the legal equivalent of a "declaration of war." As commander in chief, the president has been given a mission by Congress. By passing the AUMF, Congress has delegated to him its full war power to use in that mission. Nothing in the AUMF is directed to the courts; in fact, I have trouble finding authority for target selection anywhere in Article III. And whatever the technological changes, constitutionally I see no difference between targeting an enemy with a drone and doing the same thing with a Cruise missile or a SEAL Team. Courts simply aren't equipped to decide military tactics.
The FISA Court, on the other hand, doesn't really reach beyond Article III--judges since ancient times have issued warrants for searches and arrests, and the individuals being spied on are suspected of crimes against the United States. But I don't know of a deep-rooted tradition of common-law courts telling the shire reeve he can hunt someone down and kill him without trial.
There's yet another problem: what criteria would a "drone court" apply? In the "white paper" obtained by NBC News earlier this month, the Department of Justice says that a decision to order a strike involves three requirements: (1) the target represents "an imminent threat of violent attack"; (2) capturing the target would be "infeasible"; and (3) a lethal attack can be carried out "in a manner consistent with law of war principles." A court might be able to apply the first criterion, though just barely; but there is simply no precedent for an Article III judge balancing the prospective risks of a capture operation vs. that of a missile, or assessing the probability of "collateral damage" if the strike goes forward. We have left "the judicial power" behind altogether, and created a panel of poorly trained generals in sloppy black uniforms.