Logitech Keyboard Support
Starting with BetterTouchTool version 5.805, BetterTouchTool natively supports Logitech Keyboards (by implementing parts of Logitech's HID++ 2.0 protocol - https://github.com/Logitech/cpg-docs/tree/master/hidpp20). This allows BTT to recognize and remap even the special keys on Logitech Keyboards.
Note This is supposed to be a full alternative/replacement to the Logitech apps like Logitech Options+ or G-Hub. Actually BTT won't work correctly while any of these apps are installed.
Setup
Go to the "Keyboard Shortcuts" section in BetterTouchTool and select one of your existing shortcuts. Then click the "Add Logitech Keyboard Configuration" button:

Afterwards select the keyboard and press the apply button:

Now you should be able to record all of the special buttons and assign BTT actions to them:

Options
There are currently no further options to configure the keyboard. Please request any special keyboard feature you might need on https://community.folivora.ai