Can the customer specific item numbers be passed to Shopware via the API ?

Yes, here is an API Call example:

[
  {
    "action": "upsert",
    "entity": "product",
    "payload": [
      {
        "id": "16be39047d2f4efcb8338b960a98b733",
        "productNumber": "SW10001",

        "acrisCustomerProductNumber": [
          {
            "customerId": "1231313123113",
            "productNumber": "a23ad6684d2c4ddaa377f3dcae118fac" 
          },{
            "customerId": "50103aa350b245abac799e0812d9bc1a",
            "productNumber": "SW10098" 
          }
        ]
      }
    ]
  }
]