It's not the first bilateral gynandromorph bird found, though it.may be the first for this species.

It's less difficult than you think. Tetragametic chimeras for half-sider budgerigars have different genomes on either side, and the mechanism they occur by is through fusion of two distinct embryos with different DNA[1]

Given we know the phenotype of this Green Honeycreeper has a male half and a female half, and given how incredibly likely that the mechanism is following established ways that chimeras occur, it is not hard to say with confidence that the male and female halves of the observed bird are almost certain to have distinct genotypes. Put another way, it would be a much more radical hypothesis that "this halfsider bird with two distinct sexes would not be a chimera."

Obviously we'd ideally wish to collect DNA samples from each side of the bird

[1] https://en.wikipedia.org/wiki/Half-sider_budgerigar

I mean... we know how biology works? So...