Since this is a website like Omegle(but then a more "mature" version, as the site says it), it can also be your browser.
Please try it in a different browser and try the 64/32 bit version of the browser, since we have reports that it stopped working with a newer version of FireFox (the developers are investigating this)

If it still doesn't work, I personally don't think it's isn't FaceRig's but Kageshi's "fault" (not blaming anyone), since FaceRig is using a very commonly used DirectShow-filter to act as webcam, and (as you said yourself too) it works in programs like Skype and Omegle.

One trick you can do is try using a program like manycam and feed FaceRig through there (so Webcam > FaceRig > Manycam > browser > Kageshi )

Feeding it through manycam would cause input lag wouldn't it?

yes, but less then FaceRig itself.
If you are worries about de-sync I suggest you to also take a look at this guide if you haven't already (plus you can use some voice effects):
https://steamhost.cn/steamcommunity_com/sharedfiles/filedetails/?id=311589681