I’ve not had any extraordinary difficulties booking awards on JL merely calling AA (though iirc I’ve always used AA awards on JL for gifts to others, including two awards in F to abc from Asia, though not for myself).

The reason I think calling the Mindpearl call center in Suva, dba AA call centers Australia and New Zealand, is that Mindpearl has access to more than Sabre as their GDS.

JL has recently (March 2018) renewed their partnership with Sabre (and iirc still is involved with AXESS), so I’m not sure Mindpearl would be any better than AA agents at securing AA awards on JL. But note the information in the Wikipost about JL releases of award space.