If it is true that tablets are being taken from sites, then it is unlikely that we have facsimiles. But perhaps many of these "looted" tablets will eventually make it to new owners who will make them available for reproduction. And if these are actually several hundred thousand tablets of day-to-day life, it is hard to argue that the ultimate loss of any handful will destroy a record of the "big picture." It is more likely that the sheer number of tablets to be read will prevent us from getting to relevant data, if it exists, in a reasonable time.
How do you figure they know that they contain weather information?
IMO, lake sediments and ice cores tell a clearer picture of past weather. So...