Is there a way to create a fee that doesn't show in the store but is added to each order at check out?

Something like a "Service Fee"?

The only additional fees which work that way would be Tax rates, and those are always a percentage of each line item (product price and shipping rate have tax applied individually).

In order to add a static amount to each order, you'd need a developer to write custom application or plugin for it which would add a new item to the invoice.

Thank you.