OK. I've been doing some poking around using Fiddler (debugging proxy), Hopper (dissembler), and some other tools. I've emulated a Kobo Clara by taking the version and affiliate files latepaul provided and placing them in a .kobo folder and using the SUBST command to assign it a drive letter. Kobo desktop detected it as a Kobo and downloaded the update file. I extracted KoboRoot.tgz and grepped for the user agent string (strings libnickel.so.1.0.0 | grep AppleWebKit -C2), which contains the version. It was 4.8.11090.
So, here is the unusual part. It downloads the EXACT same file and URL as the 4.8.10956 update. No other network requests are made that are significant. The funny thing is, the KoboRoot extracted to the fake and the KoboRoot in the update zip do not match. This means that Kobo Desktop is doing some sort of in-between thing between downloading and installing. I really cannot figure out what it is.
I would really appreciate help figuring this out. If anyone wants more details, feel free to ask.
For now, here is the proper update zip I pieced together for 4.8.11090 based on the files Kobo Desktop wrote to the fake Kobo:
https://transfer.sh/Oa4Gt/kobo-update-4.8.11090.zip
I've attached the batch file to make a fake kobo.