Hello Jonathan, and thank you for posting to our community!
Yes, it is possible to view your postage costs per shipment and by store using the Complete Shipping Report or by creating a custom report.
This link will explain: How to: Export a preconfigured report from your ShippingEasy account.
And this link explains: What data is included in the Shipping Report - Complete Shipping Report?
If you prefer to create a custom report that lists only the information you need, this link will explain: How to: Create a custom report.
In regards to BigCommerce store API Keys, yes, they typically have an API Key listed in the Legacy API Settings within BigCommerce. The link below explains how locate that information under the "Entering your BigCommerce credentials into ShippingEasy" section.
Thank you and happy shipping!