What the Romans found out about was lead acetate.
They discovered that lining their wine storage containers made bad or old wine turn sweet, rather than sour. This is because the acetic acid of the vinegar reacted with the metallic lead of the lining, becoming and extremely sweet - and extremely soluble, bioavailable, and toxic - compound (nicknamed "sugar of lead"). This, far more than the metallic lead in the pipes, is currently believed to be the main source of lead-related poisoning in the Romans (especially among the upper classes, who could afford the wines in the fancy containers).
The NRA's point is that metallic lead is enormously less of a toxicity issue than water-soluble lead compounds, and that anti-gunners and anti-hunters are (in its opinion) using "junk science" claims to push for yet another piece of legislation restricting guns, ammunition, and hunting.
The obvious counter would be to bring up NON-junk-science research establishing that metallic lead from shot actually is a significant problem and quantifying the problem. That only works, of course, if such non-junk-science results exist.
That doesn't say poisoning from lead shot is NOT a problem (or not a significant one). But given the number of scientists looking for such an effect, I'd consider a lack of such papers (if, indeed, there is such a lack) would be an indicator that toxicity from shot is so low as to be buried in the noise, rather than that nobody has gotten around to documenting it.
(Now its lack in the POLITICAL DEBATE, of course, could just be a matter of the anti-lead-shot faction going with the most lurid claims as a political tactic.)