OK... now my "regenerative soil health" brain is kicking in. Obviously, the conditions are "right" for the critters to thrive right now. What DON'T they like for habitat? What works against them? What kind of plant species are found "where they aren't" vs. "where they are", etc. What are the soil conditions? What is the residue/cover conditions? You can't do much about the "climate" on the broad scale, but you might be able to do something on the "micro-climate" scale.
Reading about using tillage to destroy their earth workings.... I wonder what would happen if instead you put cattle on their workings at very high density... and let's say you actually FEED on them, by bale grazing or bale unrolling, through the winter months. That'd dramatically change the "micro-climate" and introduce heavy hoof disturbances at the same time. And then maybe even create a lane, where you would regularly run the cattle through these areas on the way to other areas? Do you have a "rainy season" when it gets muddy? Let the cattle muck these areas up badly by feeding right on them during those times. You've already lost the grass if you don't control the pest, how much harm could it do to POUND these areas by feeding on them in mud, adding organic matter and seed at the same time? And even if it's "desert" with no rainy season, still putting all that carbon down and all that hoof traffic in a concentrated way would be a really drastic environmental change from the "avoidance" situation that is happening now (because there's no grass there, there's also no animal traffic there to speak of, and dwindling organic matter in the soil).
It might even be that the "predators" that would normally function to "control" them, are absent, because of a lack of desireable habitat FOR THEM! Therefore, your "pest" critters are able to thrive undeterred. "Build it, and they will come!"
Just thinking outloud, and wondering what potential the animals could have on them... if managed differently. If the cattle pound their "habitat" severely, AND add in alot of organic matter to change the soil composition and biological component dramatically, it might just no longer be the proper "habitat" for them.
Build it (habitat that they prefer) and they will come... so change it into something that they DON'T like, and they will leave! We destroyed the habitat in conventional agriculture for most of the grassland birds... and they've all but disappeared. But if we build back their habitat, they will come back too, unless we've made them completely extinct.