Conundrum

In no way is it wrong.
They are not indigenous to that ecosystem and are harmful, they should all just be wiped-out as efficiently as possible.
On our ranch we shoot all feral cats and pigs on sight.
If you work on our property it is part of your job.
Even if you are vegan the glue that holds your shoes together, your make-up, car parts and the celluloid the film is made of when you go to the movies is made of animal parts.
Eggs are used to make your vaccinations and animals are used, and killed, to save your loved-ones every day.
I hunted for years to control populations of animals so those left would not starve or get ill due to declining space and/or predators.
I take it you don't kill the mosquito or biting fly on your arm, roaches in your home? Of course not, that would be "wrong".
It is not wrong and there is no conundrum... kill the pests.
If research is being done it should be on how to kill all of them.