The Fuji 90 mm f/2 has approximately 12.5 cm x 8.4 cm field of view (1:5 magnification) at the close focusing distance of 0.6 m. I'm not quite sure if that's enough for "head and shoulder" portraits of birds, but it might work.
The 80 mm f/2.8 macro will do 2.51 cm × 1.67 cm (1:1 magnification), which is definitely enough, while the 60 mm f/2.4 macro can reach 5 cm x 3,3 cm (1:2 magnification).
(The Zeiss 50 mm f/2.8 macro can also reach 2.51 cm x 1.67 cm, by the way.)
Your 18-55 has a maximum magnification of 1:6.6, so if that was enough for you, the 90/2 should work.
Right now, I don't think there is a native Fujinon lens with an aperture of f/2 or brighter that can focus closer than the 90/2 can. (Except for the 16/1.4 which will distort your birds badly).
I think any of the three (60, 80 or 90) could suit your needs, or you could really just add a flash or continuous light with softbox and stay with the 18-55.
If manual focus is an option, there are lots of third party lenses you could adapt or even get in FX mount.