"Starfleet was founded to seek out new life... well, there it sits!"
"The needs of the many outweigh the needs of the few, or of the one."
Which does Picard believe holds more weight? I personally think he would decide to let Tuvix live, based on the idea that "What's done is done" and that Tuvix had no choice in the matter. In my opinion, Picard would view killing Tuvix as an unacceptable solution.
Although, if it had happened on TNG, it wouldn't have been Tuvok and Neelix. Perhaps Reg and Worf? And he'd call himself Rorf. And in the end, Picard would allow him to live, but it wouldn't matter because Rorf would decide to sacrifice himself to save Reg and Worf. Rorf's last words would be, "The needs of the many outweigh the needs of the few, or of the one. Goodbye."