I have used both AF-C and AF-S with no change in the aperture operation. Focus is better in AF-S because the camera has more time to use CAF to achieve focus; only in very good light can you scrape by the AF-C.
I did a full factory reset on the X-H1 a few months ago trying to get to the bottom of this problem with no change in the operation of the 500mm PF. Had the same issues with a rented X-H2 whose settings I did not alter before I tested it with the Fringer + 500mm PF.
The request for custom firmware does make sense because the firmware Fringer has for the EF-GF adapter and the EF-FX adapter has switches that the user can set or clear for a number of lenses. They could do this for the 500mm PF if they wanted to.
IMO, Fringer has had quite a bit of difficulty getting Nikon glass to talk to Fuji cameras compared to getting Canon glass to talk to Fuji cameras. The NF-FX adapter came out much later than the EF-FX adapter, and the first one I acquired would not talk to the X-H1 at all. The second one was erratic in operation, only the last one works even at this level. Hard to escape the conclusion that the Fringer NF-FX adapters are fractionally baked.