There are several global equal area projection that you could use for the two countries : sinusoidal projection and Eckert projection are. However, because you are far from greenwich, I recommend Berhmann cylindrical equal area to avoid large distortion. Note that you could make some nice custom Eckert or Sinusoidal projection simply placing the meridian of reference at the date line or at the middle of the extent between Australia and New Zeland.

There are several global equal area projection that you could use for the two countries : sinusoidal projection and Eckert projection are. However, because you are far from greenwich, I recommend Berhmann cylindrical equal area to avoid large distortion. Note that you could make some nice custom Eckert or Sinusoidal projection simply placing the meridian of reference at the date line or at the middle of the extent between Australia and South East Asia.