I once had to dismantle a gas stove to get a sparrow out of it.
They fall asleep and topple down the flue, and once inside, the only way is down. No room to spread their wings, of course.

I had the sense to open all the windows that would, and draw curtains on all the windows that wouldn't, first. No problem, she went straight out. Through two rooms. Perhaps sparrows are more intelligent than sapsuckers. Omnivores, you see.

If you have an open flue larger than 4", perhaps some chicken wire over or in it?