Serving our Jersey suburb is a pest control company (maybe many) that will catch unwanted rodents (woodchucks, raccoons) in have-a-heart traps and allegedly transport them to public land at least ten miles away (sounds suspiciously like giving Rex to a farm family, but bear with me). Contemplating an invisible fence for a prospective new dog (alas, only blind Merlin of blessed memory could be contained by our 15-inch stone wall at the back), my sons started speculating about a have-a-heart exterminator that would catch unwanted rodents, tranquilize them and collar them.* That raised the problem of the collared pests' offspring and led us to the inevitable conclusion that responsiveness to the invisible fence would would have to be infused by genetic modification.
Ergo, the answer to this post's title is yes. And getting there (here, sort of) is the whole purpose of this post.
*We also speculated that widespread adoption would lead to pest infestation of lower-income neighborhoods abutting the served towns or neighborhoods, but that does not get us where we're going here.