I have owned maybe 10 Baccarat mushrooms. None had a date / signature cane. If you have one with them, cherish it!
As for dating the weight from the canes, my view is - 'No, not possible. Not remotely possible.' Some of the lengths of the canes that were made remained on the storage shelves, and pieces were brought out from time to time, so a weight could contain canes many years old.
I have visited Saint-Louis on a couple of occasions, and their cane store is a joy to behold: I suspect some examples are over 50 years old, and a few might even be from the 19th century.