Your chances of getting a business class Aust-USA in early April are low to zilch.
You do not state which ffp you have miles with.

As QF charge $$$ for some/many seat selections and even for business/first some people do not select seats, so seat maps have zero indication of sales.

Easter is the busiest time for flights in Australia. Thursday/Easter Friday has the most air passengers for any days of the year. Flights may sell out.

Airlines that fly USA/Canada to/from Australia (not from all airports to all airports)
- United UA (LAX SFO – SYD MEL) [Star Alliance]
- American Airlines AA (LAX - SYD) [OneWorld Alliance] (AS partner)
- Qantas QF (LAX SFO JFK DFW HNL – SYD BNE MEL) [OneWorld Alliance] (AS partner)
- Jetstar JQ (HNL – SYD BNE)
- Delta DL (LAX – SYD) (VA & AS partner) [Skyteam Alliance]
- Virgin Australia VA (LAX – SYD BNE) (DL partner)
- Hawaiian (HNL – SYD BNE via Hawaii) (AA & VA partner)
- Air New Zealand NZ ( LAX SFO via AKL) [Star Alliance]
- Fiji Airways FJ (LAX via Fiji) (AA & AS partner)
- Air Tahiti Nui (LAX last segments on codeshares) (AA & DL partner)
- Air Canada AC (YYR - SYD, BNE from Jun 2016) [Star Alliance]
And others via Asia & Middle East