There's actually precedent. Mustangs are a feral invasive species that deplaces local native species but they're protect because they're pretty. Whichever federal agency does monitor the herd and will conduct culls if they're population is getting too large and causing them to get sick.

Fascinatingly horses primarily evolved in North America but moved east while humans moved west, although there's a tribe of Native Americans in Canada who report having a traditional horse species for hundred of years even predating Spanish landing in Mexico. Unfortunately genetic tests if the remaining member of the herd isn't possible to disprove due to cross breading. There's also no example of remaining wild horses in the world, just like there's no wild cows. Part of the issue is domesticated horses getting loose and interbreeding. Honestly it's surprising Wolves have remained, although as a predator I'm sure it's easier to remove wild dogs.