Probably not. Only two things tied Clay's murder to the Phantom: the moon rock, which no one could actually prove for a while since it was out somewhere in space, and the lighter, which the Phantom held in his possession. Though showing up to court may not have been the best idea he ever had, he'd be safe if he had taken the extra step of stopping by the museum to pick up her fingerprints. By the time the lighter was questioned as evidence, they found out it was a gun. Fulbright may have skipped over the usual protocol to verifying the validity of evidence, but without anything else to counter it, it would have passed.
The problem comes before all that, though. To even get Athena's prints on it, he'd have to know Athena would be lying unconscious in the museum. If he somehow did, he'd then have to ride along the moving launch pads to the museum, pick up her prints, and swap the pads yet again to reach that ladder and escape. However, Captain Segway, hero of the people, would have noticed the pads being swapped again. He'd then have to swap the pads a third time, so Launch Pad 1 and the museum would be in the right place. Either way, we'd be able to tie the Phantom into the case, via his psyche profile, all thanks to the photo taken just before the murder.
Then, the issue becomes "why would the killer need to swap the pads again, when he could have just leapt to freedom before the pads reached the museum?" And no, "he was hesitant the first time" doesn't count. The only reason would be that he somehow knew Athena would visit the museum that day.
But that doesn't make sense in the first place, since her visit was a coincidence. Even by the slimmest chance that he somehow got Athena's prints before the murder took place, he had to have a reason to meet with Athena, ask her to hold that lighter, and take off with it. Even he couldn't have foreseen things from that far back; Clay put up unexpected resistance and got his blood on the lighter.
Therefore, Athena's prints couldn't possibly have made it onto the lighter, and this what-if scenario is impossible.
By the time the Phantom's cover was blown, the lighter was proven to be forged evidence. The only thing left to use to tie him into the case was the moon rock, so yeah, they talked all about it.
