(If that's the explanation, another interesting thing about this graph is the uptick in the early 700s, presumably due to the Moorish invasion. By 750 they seem to have been mining on a greater scale than the Romans, which is not what I'd have expected.)
The paper specifically calls out mining activity around the Rhine as having significant impact on the lead signal, with a 2-5 fold greater impact compared to Spanish sites. The authors suggest that the 700s spike in lead pollution was driven by increased activity in France and Britain (which once again has a many fold greater impact on this measurement than in Spain).
Also, doing a cursory look at sources suggests that the eastern empire relied far more on gold and bronze coinage than silver, suggesting another possible source for the lack of an Eastern Empire signature.
Poking around, it looks like there is evidence for lead mining in the Derbyshire region possibly including Roman times, but there isn't any great data floating around.
The other source the authors suggested (Merovingian mining and coin production in France at Melle) seems to have far stronger evidence for large scale production in that time period.
On the other hand, the lead's there in the data; it must have come from somewhere; so that part of the graph seems to me the most intriguingly mysterious bit.