I never tried it with FaceID specifically, but all the other red bubbles I've encountered behaved the same way: You say "not now" during the initial setup, then you get a nag/reminder in the settings app. If you tap on the reminder and say "not now" or "don't use feature XYZ" or whatever again, it goes away permanently.

Huh -- you're right. I went in to FaceID setup, immediately canceled out of it, and the bubble is gone. We'll see for how long I guess.

In my experience, it lasts between indefinitely and until the next major system update. I think, that has been the default behavior for all of Apple’s „use our services“-reminders since they started showing those.