Typically the carrier will invoice the party who is paying the freight charges for the loading of the container on the vessel as a part of the ocean freight; if the ocean freight is prepaid, the loading is included in the charge to the seller. 

If the buyer is paying the freight charges under freight collect, then the carrier will automatically include the loading in the bill to the buyer. The vessel loading charge is not separately invoiced unless there are goods that are oversized or require special handling. It is highly irregular for the carrier to separately bill the loading charge if the product is moving in a container (20', 40', 40' high cube, etc.). 

There are two possible trade terms:

FCA carrier, pier, Port of Long Beach, CA, USA, clarifying that you will pay for stevedoring charges with a supporting invoice from the port operator. See the port tariff for Long Beach at the Port of Long Beach website. This term means the ocean freight is freight collect.

CIP port of discharge, which means the seller pays the ocean fright and cargo insurance and invoices the buyer on their commercial invoice.