Thanks for posting.

When I did theater as a teenager, we had a protocol for stuff like this - intruding animals (yes, it happened), dropped items or items from the prior scene left on stage by mistake, missing props that an actor would need to interact with, etc.. When it was noticed, the actors would pause or freeze. Then the stage manager would quietly walk on to retrieve the item or fix it (if it was a brief fix) and then left the stage. Then the actors would resume the scene.

This video seemed like a more professional troupe than just some local summer stock gig. I'm surprized a stage crew member didn't come on stage to try to retrieve the kitty.

It was a Russian troupe but they didn't seem bothered by it. Also, removing the cat probably wouldn't have gone over well with the audience.