Probably included way too much information here but...
You can, but you will probably be using two different fares (econ and biz), which makes for wonky prices. I'm not sure about booking over the phone in different classes (I'm sure someone else can advise), but you could also construct the fare and buy it with an online travel agent (though you have to deal with them when things go awry).
Looking on matrix I can see QF non-stop return in March but the one-way HND-SYD in J is around $7k - takes the total to over $10k because you can only add a flexible economy ticket. You could buy two separate one-way tickets to save a few bucks. If money is no object, I guess buy this one, but given you are asking about additional fees, I think you might be thinking twice about a 10k ticket.
For other non-stop options, I can find ANA and JAL return for ~$3500 AUD in March with Economy (S/D) there and Business (P/C) back. Japanese carriers are probably a better product anyway, though you will earn less SC than QF metal (and none with NH)
Big component of the NH fare is a massive YQ fuel surcharge, but price comparable to JL.

And the QF fare I could find...
