Okay, so: as one of the poisoned, Lan Wangji would absolutely want to chase down this flappy bird and shake her of some feathers.

Buuuuuut he has no entry point in Vannozza's quarters to get the key to the hunting grounds — if someone else can magic that miracle? Or he could help with slipping in to steal borrow the keys, but he'd for sure need companions in both this and in raiding the hunting grounds. He'll be too sick to be efficient.