I rang up Qantas Health Services again, and got a different person, but this was acutally better, because he was a more "in-charge" person. So I explained my difficulties about having no control over when I got the booking information, and therefore not necessarily being able to put my medical clearance form in with the requisite 5 days of lead-time, and he was so nice and helpful; he said that when I did get the information, fax the form in as per the normal proceedure, and then give him a call (he gave me his direct line) and he would arrange for the engineering people to be there as is required.
Apparently they don't need a special plane or anything (which makes sense, since I know from my work that international aircraft are often assigned to a particular flight about a month beforehand) but they have the engineering people there to plug in the machine and make sure that it works okay before the flight.
This is such a weight off my mind! Whoopee!