As of May 19, 1986, people with missing fingers are prohibited from owning NFA weapons.
j/k. I'm sure that they make note of that while printing you and just send it in. You can't be the first person printed who has a "I lost this little guy while ......." story.
The person taking the prints will just have to enter in that finger as an exemption into the system for that finger. It's no big deal as long as the operator enters it BEFORE he begins taking your prints.
there is a spot on the hard cards for annotation of missing or disfigured digits not sure how live scan does it, my guess is they just add it to the information they send with the prints.
I had Soldier in my unit who didn't have any ridges on his finger tips. The guy was a laborer who worked outside with tools and digging ditches. Shake his and was like raw course sandpaper. He had the hardest darn time getting his clearance. They investigated him for months, they thought he got rid of his prints on purpose.