If I recall the law itself allows the President to waive the restriction. So in effect exporting cluster munitions requires Presidential approval. Whereas the approval for export of weapons is generally delegated to others in the State Department.
Ok. If that is correct then my mistake. I guess I didn’t get that detail from my initial source. That makes me curious thought, who is being granted the approval from the president? Isn’t the executive branch of the goverment who desides for the export in the first place?
I think this page here will help you out with all that kind of information:
https://www.state.gov/u-s-arms-sales-and-defense-trade
It looks like the Bureau of Political-Military Affairs handle approvals in the State Department.