For some reason installing FieldLink appeared to have no effect on the error either.
So I decided to fire up Process Monitor from Sysinternals to gain some insight into this "License.xml" file that Siteworks uses.
It appears that Siteworks first checks the registry (HKLM\SOFTWARE\WOW6432Node\Trimble Mobile\Files\License) for the location of the license file, which on my machine ended up being "C:\ProgramData\Trimble\Common\license.xml" (not in "Program Files (x86)" mentioned above) - maybe this can vary from machine to machine.
With the .xml file located, it was a simple matter to rename it as described earlier in this thread, et voila the license error message is gone.
Siteworks still reads the registry key, attempts to read the license file which returns NAME_NOT_FOUND, and then proceeds to display a message informing me that the application is not licensed for this controller and that it will run in emulation mode.
Success!
I'm not sure how my system came to have a license file that wouldn't work in the first place, but it seems Siteworks behaves properly only with either no license file, or a license file with a valid license - my copy has "UNLICENSED" in the serial number field, and product name is identified as "RFC Production v1.0" (not sure if this helps trace file origin).
(partial screen-capture, omits part of the file)
Thanks to @Steve DiBenedetto for your help on this!