In the meantime, I have edited my previous post with hyperlinks to temporarily hosted files, I will attach them again here in the Community as soon as things have been figured out.
Since I did a fresh install of both the IDE and the the board support I am also running short on further ideas 😞 I also selected the "NodeMCU 1.0 (ESP-12E Module)" which compiles successfully (but I only have the Adafruit Feather HUZZAH to confirm that it actually runs as expected).
The generic release package on our website is the official package for BSEC. The BSEC Arduino Library is a wrapper (intended) to ease the use/integration of the BSEC library in the Arduino environment. The current version of the BSEC Arduino Library is up-to-date to run/support the current latest official BSEC release package (v126.96.36.199), thefore these files should be identical with no need to manually replace anything manually.