Being an APS-C lens (or mirrorless) won't help much with that focal lenght and maximum aperture. I think that has been discussed somewhere on the forum already, but I'm in the mood for it now, so everyone who does not want to read it again forgive me (or quit now):
A very big chunk of the weight of fast primes are the lenses' front elements. A 200/2 lens always needs a front element with at least 100 mm (+ a bit) diameter, no matter what the format of the sensor/film in the camera is. If you look at the AF-S Nikkor 200 MM 1:2G ED VRII, for example, you can see that behind the thin, flat protective glass element in the front, there are 3 giant, thick lens elements (2x ED convex,1x normal concave), which can't be reduced in size. Those three lens elements alone are probably responsible for 1/3 of the nikkor's weight. (They should be about 10, 20 & 15 mm thick and have a diameter of about 100 mm, roughly estimated)
They have to remain their size because of the way light travels through the lens. There's a nice image on wikipedia that I'd like to borrow for this explanation. I know it's a microscope and not the nikkor's structure, but nearly every (photographic) lens works the same way, and due to the microscopic nature, the object plane is close to the front element which helps a lot with visualising the explanation:
(source: https://upload.wikim...al_path.svg.png )
Take a look at the lower marginal ray (green). This describes the path of the light from the lowest outermost point in a real scene the camera can "see". You can see that every point (in the real scene) reflects light in every direction. All the light that is reflected towards the front element of the lens, up to a certain point where the incident angle becomes too flat, is now collected by the front element, thrown somehow through the whole lens and focused back to one single point on the sensor.
This means, if you reduce the size of the front elements, you gather less light for each single pixel instead of cropping the frame, which makes the lens slower, no matter how big the aperture is.
You can, however, reduce the size of some lens elements deeper in the lens. The thing is, they are a lot (!) smaller already, so you don't save much weight by that. (again, the picture does not represent the 200/2s structure)
I don't remember if the Canikon 200/2 are real telephoto designs, but if they are not Fujifilm could spare some weight by building their's as one and reduce the lenght of the lens body (and compensate for their shorter flange distance, which would have required the lens to be longer than the Canikons). But again, not much to be saved there, and it would probably demands more glass, which compensates for the lighter lens body.
With all that said, I think Fujifilm could make their APS-C 200/2 a bit lighter than the FF 200/2s, but I don't think they can shave off more than those 300 - 400 g I mentioned earlier. (well, of course I hope I'm wrong).
Fuji's own 100-400 is an example of this: 1.37 kg vs 1,5 kg for the FF counterparts.
The often praised APS-C size advantage comes into play when you compare an APS-C system with a FF system that gives you the same angle of view. (200 mm vs 300 mm, and so on). But most people, although realizing and acknowledging the DOF difference between a 200/2.8 on APS-C and a 300/2.8 on FF, don't take into account that the ISO rating already compensates the loss of light every pixel has in the APS-C system vs. the FF system. That's good, because this way, ISO, shutter speed and aperture can be used globally and interchangeable, and that is also important for flash photography. But the size and weight saving of an APS-C system is bought by higher shot noise (worse signal-noise-ratio). This can be partly compensated by faster lenses, like Fujifilm did in the past (and only up to the limit of the full well capacity, limiting dynamic range). But then, your APS-C lenses are just shorter and fatter than your FF lenses. Knowing that sensor developement marches on with great steps, I still think Fujifilm did the right thing with chosing APS-C. But one can't just ignore the fact that a FF sensor of the same generation will always offer better noise performance and a larger full well capacity.
Uuh.. well, I think I started to blabber in the last paragraph. Please pardon me, it's late where I live.