It has come up now, so removing the issue of which store to pick up at, is this a possible solution:
In the product set up, instead of using weight, since they are not using FexX, UPS or USPS, set a price for shipping. Then in Options/Advanced Options, set up a selection for FREE PICK-UP or the Shipping cost for that product.
I suspect that the largest store will have the inventory, but I have to confirm that. But in the FREE PICK UP option, it could indicate which store to select. More information to get from customer.
Or am I overthinking this and in the shipping rates, there can be an option for Free Pickup?