When using BOPIS with eBay there are issues with eBay that need to be know, so then can best decide how to run BOPIS:
(Info valid at 22 July 2019)

Remove out of stock items
This is the safest option, and means that when an items online stock is 0, then the item is removed, and not available for BOPIS.
This will remove it from search results, and not make it purchasable (which currently happens when its changed to out of stock item).
EBaySetRemovedItemsToZeroQuantity = false
EBayListOutOfStockItems = false

Set to zero out of stock items
This means that when an items online stock is 0, it will still be available for purchase by BOPIS.
Bugs in eBay mean that it is still purchasable (but might cause error after purchase), and is not currently recommended.  If using this it is recommended to use a rule to set the Shipping Policy to use a pickup only shipping policy.
EBaySetRemovedItemsToZeroQuantity = true
EBayListOutOfStockItems = false

List out of stock items (BOPIS Only)
This method is used for BOPIS only items, it will still be available for purchase by BOPIS.
If using this it is recommended to use a rule to set the Shipping Policy to use a pickup only shipping policy.
EBaySetRemovedItemsToZeroQuantity = true
EBayListOutOfStockItems = true