Props to Ace and VarlosZ on well-adjusted cat ownership.
My problem is not as much about whether killing the woodpecker is wrong, but whether the cat dines on the woodpecker, post-hunt. I understand the emotional distress, I anthropomorphically sympathize with a limp woodpecker far more than a cricket carried around in my cat’s teeth- and I don’t even resent the cat for not having cricket on toast points later (blech, who wants to eat a cricket anyway.) The issue is whether the woodpecker has more value than the cricket and, if true, whether that warrants action on Ace’s part.
If you think that Merengue is going to dramatically improve his serial woodpeckering, you may want to train him out of it for your own mental well-being. However, unless there are an overwhelming number of woodpeckers felled at the hands of your remorseless cat, this is just another part of natural selection. I feel like your cat is not at such a ridiculous advantage that you will have to level the playing field