If you can not live without the critter move to where it is not an invasive species and you can have it.
Invasive plants are a big problem in my area. English Ivy may make your house look cool but in the wild it smothers trees. (harmoniously of course) I know you have the same problem in your area with kudzu.
That is not even considering all the little nasties they can bring in with them.
English ivy, kudzu, Asian wisteria, privet. At least the wisteria has a beautiful flower.