Classified as a category 4 threat, the Elib is somewhere between a vampire lord and Godzilla for limits of escalation. Yes, I would consider using a nuclear device to kill just one. I am that sure that I want it dead.
A town has a festival every year in the dead of winter. The festival of birds. The towns folk dress in feathery robes and beaked masks and dance and frolic in the belief that the changes of the seasons are controlled by the presence of birds. According to tradition, the festival tricks Spring into coming early because the birds have returned.