Lyncher ALWAYS claims cop in this setup, because a nk aimed at them is a nightkill not aimed at their target, and they REALLY don't want the scumteam to figure out who their target is.
If you back down and let the night phase happen, either A. someone other than cop claim dies, nl basically confgoon. B. no one dies, nl basically conf lyncher, and (more importantly) lynched without outing the cop. C. There's a slight chance that no one dies and nl is a goon, which means that nl's partner reshoots the following night, either killing the lyncher or killing the lyncher's target AND the lyncher. Being the person to back down there makes it super unlikely you eat the bullet, so you've basically guaranteed you have 2 nights with no one on your ass to investigate and town is basically in autowin as long as you don't get run up on D1 after the cop claim.
Of course, nl has to claim cop in the first place to make that happen, but.... That's why Clemency is better than everyone else.