After receiving a few more logs, it seems like for some reason macOS is intentionally freezing the BetterTouchTool app on some machines until a user action targeted to the app is recognized. I don't know why it would do that, maybe it's a way to save resources and maybe it only does it in specific settings.
I'm trying some ways to prevent that in version 4.166 alpha, would be great if you could test that and see whether the issue still occurs. The clamshell command mentioned above might still be helpful because it doesn't seem to be the same issue causing this for all users.
If it still occurs afterwards, you could try to disable app sleep system wide to see whether it's the cause of the issue:
defaults write NSGlobalDomain NSAppSleepDisabled -bool YES
(This probably requires a macOS restart - you can change it back to NO at any point)