Wages would go lower but goods will be higher. If an immigrant worker produces more than he consumes, the whole is richer, no way to cut that into an unfavorable calculation.
We get more goods with fewer and fewer people in assembly and we're no better off for it as far as structural inequality is concerned. I would check the assumption that a cheap immigrant worker would increase productivity at all.
Again, if the immigrant produces more than he consumes, the whole is richer. The arguments against that calculation would be that you dont like who benefts on that, but the whole is better.