I have owned a few Fuji bodies and lenses, the last body being the XT1. I changed back to Nikon for the longer lenses available and intially bought a D750, but due to a fault the camera had to be sent back and as it was close to Christmas I bought a D7100 and 300mm f4 AFS prime lens. For comparison, at that time, the Fuji wasn't any where near it in AF terms but the Fuji ISO was probably better in JPEG mode. I know Fuji have brought out new FW updates so it maybe better now. Also a point of note, the Fuji lenses are excellent in quality.
The D7100 never failed to lock AF even with a 1.4TC for the day light winter months I used it, it just done the job. As for your 55-200mm lens it's only at f3.5 at the short end and goes to f4.8 at the tele end, which for me isn't a very fast lens at 200mm. I have a Tamron 70-300mm VC lens on my D750 now and it's at f4 @ 70mm and f5.6 @ 300mm.
If the XE-2 is working for you keep at it and with more experience your keeper rate will get better, as for changing to an XT1, the AF is supposed to be better but is it a worth while upgrade, I'm not sure if staying mirrorless.
If you were to change systems I would look at the Nikon D7200 and a Sigma or Tamron 70-200mm f2.8 (or Nikon version if you had the cash) or 300mm f4 AFS lens which could be probably bought cheaper than the Fuji XT1 and 55-200mm lens if purchased from somewhere like HDEW. Also, with the Nikon cameras these are now 24mp over the 16mp with the Fuji which will allow for more cropping if needed.