If you won't open the windows, don't crate Bijou in the kitchen while cleaning the oven. That would be bad.

I'd say go ahead and clean your oven, but you really should at least crack windows for ventilation. If your bedroom is on the other side of the house, just shut the door with Bijou in the room with you and bundle up. It won't take that long to rewarm the house once the windows are closed.