I doubt it - they've never offered it as an option for connecting flights from PER, it's always been the red-eye flight out to BNE, SYD or MEL, or the 5.30am flight out that morning to SYD to connect with VA1 in the afternoon (which will no longer be an option when VA1 commences the earlier departure time from late October).
As catching the red-eye, hanging around the airport for anywhere between 6-8 hours and then getting on the long haul flight was such a killer, for our current trip we wondered whether we could do what you suggested - catch an afternoon flight from PER to the east coast, stay overnight and connect to the LAX flight the next day.
As you mentioned it's less than 24 hours so wouldn't (shouldn't!) be classed as a stopover. I contacted VA and was told that we couldn't opt for an earlier flight as a "connecting" flight, what we wanted to do would be considered a multi-city booking and would have to be done through the GCC.
Maybe they might consider it for MEL though as you don't have the "overnight" flights we have from PER.