V5.763 contains the very first implementation of full native Logitech Mouse support that might work. (It might also not, I only tested it with a limited set of mice and configurations).
The most important part is the Logitech Configuration Object in BTT's Normal Mouse Section. You need to add one such for every Logitech Mouse you want to use in BTT. Afterwards you should be able to configure your buttons like for other mice.
It might currently require a BTT restart after adding and setting up a Logitech Mouse Configuration
I'll now start testing with all the Logitech Mice I can find.
If you want to dynamically change e.g. the SmartShift / Ratchet mode, there is a predefined action for this now.