The opposing side can use Big Bad Voodoo or Molecule as soon as pistol shots are fired, and look for the source. At that point it's pistol vs. whatever primary you have on hand. And if he goes invisible by switching over to knife, won't the bird have a blood trail like vagabonds do when they get hit for you to follow?
Also, if a bird is camping, you can still use medusa's gaze or entangling roots to identify if they're camping (even if they can glide out of it, it still targets them). Simply shoot in that general vicinity as soon as you activate it to get a blood trail going.
Another way could be using flaming stike and radian's ultimate would give you a target (in the case of the former), or a path to the target (in the case of the latter).
Lastly, use smoke bombs / flashbangs like you would normally (nothing in rules against using hunter smokes at objectives, just wards?) This can give you cover as well as damage.