I don't know what to suggest; I'd probably traumatize her with the crate, rather than lose her outside. As the choice, that seems the lesser of two evils.

I'm with wanderer though, and maybe you can find a bigger crate.

Less convinced by the idea of trying to convince the cleaners not to let her out when they do the windows. That only works in the case of my fur child, Smokey, because the flat door is never open for more than a few seconds, she's not that keen on going outside to start off with and she can be easily convinced not to bother. Plus, she doesn't like my cleaner so won't go near her, even go get out.

Borrow a bigger crate would be my vote.