"Invasive" is a made up human concept. The question is whether there are species that now rely on these as a food source. The entire natural world is a hodge-podge of "invasive" species coming from some nearby land, disrupting the ecosystem, then the ecosystem evolving around it – typically a gradual process over thousands of years.
If it took a couple hundred years for this species to proliferate, it's absolutely plausible that eradicating it in a few years will have surprising second- and third-order effects.
“Invasive” is a useful concept to describe species that spent millions (sometimes hundreds of millions) of years evolving in one environment and then moving to a new one in the last few thousand years.
Do you understand that time scales are important here?
There are also species that have invaded areas that are totally impossible for them to reach without seafaring or airfaring human civilizations to carry them.
All terms are made up. Some are useful and some are not. This one is useful.
The usefulness of the term to this conversation is as described: is this species systemically critical to its current ("invaded") ecosystem?
Merely calling it invasive does not answer that question, ergo it is not a useful term in this conversation.
And yes, timescales do matter. A species can become critical to its new ("invaded") ecosystem within a few hundred years such that removing it within a few years will cause catastrophe.