I'm not aware of any school buses that will transport your child from Drummoyne to Mosman. You could check with the school to see if there is any pick up in your area. I doubt it though - buses that circulate near Mosman have probably never been south of the river - over the Harbour Bridge is foreign territory - and they won't have the appropriate passport.
You will have to move to Mosman - or have your child educated in some unpretentious western suburbs school. There's probably a pickup bus for PLC Croydon. I've seen it in Haberfield.