I'd argue there is one native grazer to Hawaii.
Cattle do not and cannot replicate the Nene's grazing, neither in what whey eat or what they excrete. And even then, I doubt that the pasture they are on is native grass. Grasses so easily turn invasive and cause massive problems. A big part of the wildfire issues in the western US is because all our native grasses are taken over by cheat grass and red brome instead. Hawaii has issues with multiple invasive grasses/forage. How linked they are to livestock I cannot say myself, but I wouldn't doubt it.
The Nene is a Canada goose softened by 500,000 years of island life, and they deserve being recognized as the native large herbivore, just as kangaroos, giant tortoises, and other isolated island weirdos.
There are currently 2,500 Nene remaining. It is recovering, but nowhere near the 25,000 Nene that were present when Cook arrived and colonization began. (But there were also several species of other large grazing ducks/geese in Hawaii that went extinct shortly after the first human settlement, so who knows what their numbers once were).
But I agree - large ungulate farming should not be happening on island ecosystems if at all possible. There's good reasons that Polynesian people primarily used fowl (chickens, and obviously some of those native grazing waterfowl) as well as pigs (which in Hawaii, were let to go feral and self-sustain by the native Hawaiians) and dogs as their livestock. All live pretty well on the scraps provided by humans without needing to graze.
Even if they are not out grazing themselves, the food for them is either going to have to be grown locally or imported at a steep cost. I don't know how viable the plant ag products of Hawaii are for cattle feed, other than sugarcane bagasse, but I don't think many of them are suitable.
Cattle just aren't suitable for tropical islands. But there's lots of places an operation like this could exist without being nearly so dangerous to the ecosystem. The best option is a place where wild cattle or bison species were native and no longer exist. Bonus points if he doesn't want to cull by doing it in a place with native predators to do the job for him. A little dark, but the wolves of Montana would love being next door to this operation, especially with inattentive dairy cows not defending their calves.
I don't think death by sacrifice to predators is good welfare from the livestock's standpoint, but neither is doing no population control at all.