This happens on macOS 13.5.1 with BTT version 4.214 on my 2023 16" MBP. All other triggers like 4 finger swipes continue to work. I have to restart BTT to fix 3 finger clicking.
Enabling restart after wake from sleep and process priority helper do not seem to help.
The only 3 finger system gestures I'm using is for swiping between pages.
I originally saw the issue after updating from BTT 3.952 to 4.x on my 2019 16" MBP. As a workaround, I downgraded to the older version, but 3.952 has different issues on my newer Mac.