I thought I had a solution, but maybe not!! I have both the LP preferences on the Chrome browser and the LP app preferences set to log off quickly. When I want to use LP to log into a website, the login screen from the Chrome LP extension comes up, but since I have not enabled "Remember Password", it sits there waiting. I then start the LP app, which does have Remember Password enabled, but also Use Touch ID. So the app asks me to use the fingerprint reader on the MBP, and when I touch the reader, it logs me in to both the app and Chrome.
So that seemed like a workable solution. But then I realized that once I tell the app to login and it brings up the dialog box that instructs me to use the fingerprint reader, the LP extension in the Chrome browser immediately logs me in (without a password or Touch ID). This is bizarre!