Is there an upstairs or far away room with a door you could carry the cage to?
I would do that. That way, the bird will still be safely away from the kitchen but still be in the house. If the only room with a door is right by the kitchen, I would plug up the door cracks with towels to prevent any microscopic residual fumes from creeping in. But I'm paranoid like that.
As long as you don't take him out of the room until there is no trace of the chemical and you have rinsed everything well, I'm sure he'll be fine.
Also, if at all possible, I would open the windows while you work to diffuse the chemical fumes.
This is generally the strategy I use whenever there's smoke from burnt oil in the kitchen, I'm using a non-stick pan (which is rarely, I only use it to make Crepes once in a blue moon) or if we are painting, or anything similar.
Hope this helps and best of luck with your stubborn drain, I do hate it when they get clogged!