I don't want to be a wowser, but you have a number of obstacles which you have identified yourself - leaving Australia to get to Europe in peak holiday season; travel from Lima to Mexico; and getting two award tickets (presumably in J). I would start looking at options out of PER via KUL or CMB. From Lima I would look at US airlines to see whether you can transit there for onward travel to MEX.
Also note that you are asking about RTW and this thread is for OWA - best to keep them separate to avoid confusion.
