Considering one of the judges who would have tried the case, John Underwood, was a die-hard Union supporter and the other judge who would have tried the case, Chief Justice Salmon P. Chase, would later write the majority opinion in the Texas v. White decision ruling the Southern acts of unilateral secession unconstitutional, I believe the chances of a Davis trial being dismissed on the those grounds hovered between zilch point squat and none. Also, the Davis trial would have been a criminal trial. Criminal trials do not decide what is Constituional and what is not. Only the Supreme Court can do that.
Since the trial would have been held in Virginia there was a very real fear that some people on the jury would not have voted guilty under any circumstances. But I believe that a jury could have been found who would have convicted. Jury packing is not a 21st Century discovery.
He did so by fiat, by the way. One of those emanations from the penumbra.