I can only give a rough outline of the steps through this forum. If you would like us to help you do this directly you can sign up for a premium support incident and we can do screen sharing and walk you through it directly.
I was assuming your shipping table had weights, regions, descriptions, and costs. If it is as simple as just having types and costs and the costs are the same regardless of region and number of items/weight in the cart then you could:
1) add a list/menu to the checkout page and populate it from the recordset using the "dynamic" button in the list properties window.
2) Set the text of the list to the Description, and the value to the cost
3) Use the set session value server behavior on the confirm page to store the submitted cost from the list into a session variable. (you may want to add the checkout form to the bindings so that you can bind the values using the lightning bolt)
4) Update your shipping rule to get the value of the shipping cost from the session variable you stored.