What's the Buddhist perspective on permanently eradicating bedbugs? (Extinction) On one hand, killing... 

On the other hand, if there are no bedbugs (extinct status) no one can be reborn as a bedbug. The species causes more suffering then good. 

Perhaps it would be worth it to be reborn in the hell realm for a life or two as a sacrifice to better the universe in such a way... 
Or perhaps the very thought is in grave error...

Say a man were to carry out the mission of bringing bedbugs to extinction, and then take the precept and never kill another thing for his entire remaining life. 

What is the Buddhist perspective on this?

Would such a man be responsible for removing the suffering caused by bedbugs if he had not taken on his mission?
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.