Hello, long-time reader / lurker, first-time poster; after years of reading this site, I finally joined just to respond to this thread.
I'm also having the same issue of my new Logitech MX Keys waking my MacBook Pro running Monterey from sleep soon after beginning it, and I've found it also prevents screensaver from automatically beginning as well, which is especially problematic for me since my LG 5K2K monitor is prone to image retention (screensaver prevents and refreshes ghosting when not in use). I'm finding this behavior takes place whether the keyboard is switched on or not, which points to a software / driver issue. I've tried uninstalling Logi Options, but still have Options+ installed, so I'm unsure if this will fix the problem.
I've responded to the only support thread about this I could find in the Logitech support forum as well:
I am using MX Master and Craft keyboard with Macbook Air M1. Ever since I updated to MacOS Monterey my Mac will wake up immediately after it goes to sleep. This unwanted behaviour will only go away...
support.logi.com