What's the Buddhist perspective on permanently eradicating bedbugs? (Extinction)
Being in a position of not having to kill is better.  The idea of getting rid of species that cause more harm (or bad karma to themselves) is a product of pondering mind, and definitely not Buddha's teaching.  

To answer your question, Buddha did not speak of any situations where killing another species was worth anything.  

However, if you search "kill" in sutta, Buddha used "killing" in a context of anger.  Something like "those who killed anger live in peace" , and other contexts as well, but never in a context of taking lives or other species.  Your hypothetical doesn't lead to enlightenment or cessation.