Depending on the answer to that question, yeah, it seems fair to think of manual labor as a way to offset the cost (though I doubt the taxpayer is seeing the benefits of those offset costs)

I feel like the answer here is to kill private prisons entirely (they should never have been a thing), still have prisoners do labor, but have that labor specifically be sorts of labor that either benefit the public directly (road cleanup, for example) or reduce the costs of imprisoning them (lots of labor to be done to feed and clothe the inmates, lots of facility maintenance as well, for example).