The play to probably contain the most birds would be At the Wake of the Earth-Shaker, the end of the Selkie cycle. That's a notable play for its own reasons.
In those very niche circles where Rutherford's works receive performance, Earth-Shaker is infamous for its reputation as unperformable. The script includes the direction "[ The larks eat the lead actress's innards. At this point, the audience screams and runs out. Ambulances are called. By the time the police arrive, no trace is left of the actress or of the larks. ]" The play has one single setting, and it makes no reference to any meta-play or audience, police, or ambulances elsewhere; this is referring to literal events. And the play goes on after this with no further break from convention! So for obvious reasons performances skip this part.
Rutherford sometimes had a sense of humour. He liked to describe Earth-Shaker simply as "the curse." In interviews he would pretend he wrote no such scene, in fact that he wrote no larks into the play whatsoever.