Extending or Customizing ShopifyProductHandle Product Selector

Would like the ability to extend, customize, or configure the product selector generated by the ShopifyProductHandle field. Currently, the product selector shows a list of products by product title and allows you to search by title. However, it would be nice if there were some ability to be able to either customize or configure this selector so that we could view and search other properties on the product.

For example, in our use case, we have multiple product entries in Shopify that share the same product title. This is due to having multiple sizes of each product and instead of size options on a single product entry, we separated them out into separate entries per size for various technical reasons. This results in having multiple entries in the product selector say the same thing, despite each technically being different products. Ideally, it would be beneficial if we could also include the item number in this view (less important) and even have the ability to search by item number (more important).