Hi VLocke,
It sounds like you are close, however it depends on how you have setup the Class Dropdown field in your Customer table. You should be good if added the new field by going to Customize->Tables/Fields, then created a new Dropdown Field Name (in my test I called it ClassCustomer), and when you click "Add Field", you used the option "Choose from an Existing Table", which points to the original "Class" table that is being used in the Invoice Screen.
If you have set it up this way, then all you would need to do from the Invoice Screen is add a "Retreive Value from Table" action to the "Customer" Field. If you go to Edit the Customer Field, in Step 2of6, Look at the Event "Text Change", this is where all the fields get populated when you select a particular Customer. Refer to my screenshot below for how I added the "Retrieve Value from Table" action for Class, then I placed it close to the other "Retreive Value from Table" Actions already listed.
Once you have done this, your screen should now be loading the Class Value you have stored from the EditCustomer Screen. Let me know if this helps or if you need any further help with this.
-Ben