Shopify "Local pickup" make single item available at all locations

A client has multiple locations. We’re using the shopify local pickup for collection (no other delivery options available, or local delivery, just pickup) - they are able to move stock between stores.

At checkout, the only local pickup available to the customer is the store with the stock - we don’t mind which store is chosen as we can move the stock (high price tag order items).

Is this possible with local pickup, or do we need to use a 3rd party app?

Rather than using the “local pickup”, which takes stock from stores and limits customer collection options when stock can be moved between stores. Is it possible to use the normal shipping rates and setup a free rate titled “collection from store X”, and another “collection from store Y”… etc etc then stock is allocated from either store?

Are there any issues with doing this?