Okay, I think I'm getting a clearer picture of your issue. You don't want to kill a fish, because you have some moral qualms about it.
This sounds to me like the Trolley Problem. Here's how it goes: A trolley is running out of control down a track. In its path are five people who have been tied to the track. Fortunately, you could flip a switch, which will lead the trolley down a different track to safety. Unfortunately, there is a single person tied to that track. Should you flip the switch or do nothing?
You don't want to kill the snakehead when you have to be an active participant, but you are okay with leaving snakeheads alone in waterways where they will outcompete and, ultimately, lead to the deaths of native species, but because you are not an active participant in these deaths, you don't feel responsible.
You're responsible either way. Choose the lesser of two evils.