The cause of this issue could be the incorrect permission to integrated Lambda backend with your API. So you need to use a user with proper access and use API Gateway console to add permission to integrated Lambda backend. Please see detailed steps below.

(1) Go to API Gateway console.

(2) Select target API Gateway and resource method.

(3) Click “Integration Request” card.

(4) Check “Lambda Function” for Integration type.

(5) Click pen icon to edit “Lambda Region”. Click check icon upon finish editing.

(6) Click pen icon to edit “Lambda Function”. Click check icon upon finish editing.

(7) Click “OK” button on pop up window.  API Gateway console will add permission for you.

You can try another test after the deployment. It should work.