Ah, I think I might have identified the issue.
It seems like on Catalina apps require the "Screen Recording" permission to read window titles (which BTT needs to identify the windows it is moving). This is stupid because it doesn't record the screen, but I assume that's the issue.
Could you check whether enabling BTT in System Preferences => Security & Privacy => Privacy => Screen Recording fixes the issue? If so I'll add a warning popup to BTT.