"Price Books" and Price Modifiers (Discounts)

This is related to "Price Book" functionality (Advanced Pricing).

Our goal is to be able to create "Price Books" for different Customer Ship to Site Use (Price Lists are different for same Customer depending on Ship To Sites)

The issue we are facing is that w can create the Price Book for the Ship To Sites, but the published "Price Book" only records the Prices in the Price List attached to the Customer Ship To Site.

it does not reflect the Modifiers that apply to the different Items in the Price List. Two modifiers are DFFs in the Customer Record mapped in Advanced Pricing using the Attribute mapping functionality.