Congress decided it wanted to prohibit [child labor], so it passed a law—no more child labor. The Supreme Court heard a challenge to that and the Supreme Court decided a case in 1918 called Hammer v. Dagenhardt. In that case, the Supreme Court acknowledged something very interesting — that, as reprehensible as child labor is, and as much as it ought to be abandoned — that’s something that has to be done by state legislators, not by Members of Congress.This decision was later unanimously overturned by the US Supreme Court in the 1941 case United States v. Darby Lumber Co.
Even assuming that Lee is correct in that production is an inherently local activity beyond the reach of the federal regulatory powers provided by the Commerce Clause (and he is not), the federal government would still be able to pass and enforce heavy regulations on the sale of goods manufactured by child labor to the state in which those goods are produced, effectively removing most of the benefits of employing that exploitative labor in the first place.
I live in Utah and have lived here for almost my entire life. I really disliked Bob Bennett, but then I started hearing about Lee's positions when he was competing for the Republican nomination against Bennett last year. I would've never thought I'd feel this way up to that point, but these days, I miss Bob Bennett.