I don't think I would use it without some ventilation for your health as well as the pets, and I would certainly make sure Bijou wasn't in there while it was working.

Can you not get a cleaner that doesn't take so long to work? That way you could just keep the pets out of room plus have the window open without too much bother.
The one I used the other week cleaned well (my oven badly needed cleaning too )and the maximum time you could leave it on was two hours, in fact there a warning was printed on not to leave overnight. I don't know if you can get the one I used over there - Mr Muscle Oven Cleaner - but maybe there is something similar.
Chris