The person paying may have agreed to a cabin higher than economy, but has the host actually agreed to business class?

Qantas offers premium economy from US-Australia , for example

If business class has been approved, then Qantas's business class does have horizontal flat beds .

I wouldn't take CX since the extra 2000 miles is too much

For Canada-Australia, I wouldn't be recommending the Qantas flight out of Dallas due to the extra stop. I would go for YYZ-LAX-SYD if it was me