It is never going to be a good idea to exterminate a species entirely. I think it is addressed easily by utilitarian thinking on biodiversity, removing the last occupant of a given niche endangers the system as a whole. And any given species might have been the last in its niche to survive. It is a low-to-moderate risk of a very high cost, and just not worth it.
But I think this is reinforced more powerfully by a less intelligence-focused framing of Kant's duty not to kill. Genetic niches produce goals in much the same way human autonomy produces goals. Eradicating a species removes something that has a goal in the larger scheme of things that you probably do not comprehend. An individual horse may be a means, but the entire species of horses should still be an end-in-itself. (It is an interesting question where in the middle we pass over the gap. But we don't need an answer here.)
On the other hand, if it is invasive, the species foreign and strong, so it is probably thriving elsewhere, and this is not really the issue. (This is not natural selection, it is unnatural for the species to be here to begin with.)
Still, eradicating a colony of ants may be the ant-equivalent of a killing (in a way that just killing off large numbers of ants for convenience is not) because the colony represents their sole genetic source, a piece of their identity, and their genetic trust the same way your body does for you.
At the same time, in order to be invasive, a species also has to be threatening the local ecosystem. Just being foreign does not make a species invasive, it has to endanger the species whose niches it is filling. And it has to have a genetic advantage acquired from the change in environment that allows it to do so -- tasting bad to local predators, going unnoticed as potential prey, being larger and stronger because the old environment was more hostile, etc.
Killing something that is in the process of killing other things of comparable worth via inappropriate leverage is analogous to taking down an crazed man who is waving a gun at a crowd, especially if you just brought him there, not knowing he would go off. (His craziness makes him as innocent as your ants are. His gun represents their unfair genetic advantage. Our gun represents our many unfair genetic advantages. Humans or human land alterations are almost always the culprits that bring invasive species to their new environments.)
This kind of killing in defense of others, even if your target is innocent because he is unaware of what he is doing, is defensible on many bases, from Kant to Mill: He might kill, and if he is disoriented enough, he would probably not consider himself guilty if he did so. Also the aftermath would be a disaster on a few different fronts. If you indirectly caused this confrontation, you cannot just walk away from it and let him remain a threat.
If such a killing is the only thing that is going to work, you may need to do it carefully, but it should happen.