As I understand it it's pretty controversial to promote fishing invasive species, because the concern is it could motivate people to deliberately cause the invasive species to spread.
Here in Sweden we have a invasive species called signal crayfish that carries the Crayfish plague. People do eat the signal crayfish and I have only seen encouragement to promote aggressive fishing of it. The voices of concern is against illegal implantation, but those have (to what I have heard) been mostly of a cautionary nature about people who unknowingly spread the invasive type into lakes which so far has been spared.
The European crayfish is seen as more traditional correct so the beside getting devastated from the plague it is also rather heavily fished. Getting people to stop and only go after the invasive one would be a huge win, but as the trend is currently it look like the European crayfish is going to get extinct. A very clear example where a change in what kind of species we buy would have a positive effect rather than a stop.
