I can think of dozen other things they did not account for. They are simply measuring the wrong thing.
Did they consider the timing of soda drinking? During lunch at school (more likely cans), or
while watching TV (more likely bottles). There are studies that correlate drinking at lunchtime
with weight issues.
How about 2L bottles being used by other member of the family and visitors, making it difficult to estimate the caloric intake of soda for any family member. Caloric intake is very easy to get wrong unless you perform complicated
measurements. Counting calories according to what you think you ate is only slightly better than guessing.
Some people forget to add up the calories of the huge amount of calorie-loaded sauce they add to the salad,
or the obscene amounts of oil and butter in almost everything they eat.
So maybe the lack of BPA is correlated with overestimation of caloric intake for the drinkers of L2 bottles?
Also, obese people like to drink soda during lunch (canned). If they somewhat care for their body they hope that
drinking sugar free soda will save them. So they are more likely to get a lot of BPA without additional calories than
people with a healthy weight. So, maybe they are obese because all those artificial sweeteners? Did they check
the urine for that?
If I really wanted to "prove" that artificial sweeteners are the cause for obesity, I would probably would find a way which
would look as convincing as this study. I would also be able to "prove" that people who eat a lot of quinoa are more likely to get smallpox, despite vaccination.
[hint: the link is vegetarians/new-age/vaccination level of friends]
I don't say that they did it, they almost certainly did not, but cooking-up wanted results is significantly easier than
detecting it in a review. Similarly, if the distance between the alleged cause and effect is too big, as in this case,
it is almost impossible to rule-out causation!=correlation errors.