When this happens to me, I add another node to my schema.
So for example: under the repeating node, I would have a field for the ID, and a field for the DisplayName. Obviously the dropdown stores the ID and displays the displayName, but storing these separately means that I can use them later in other places in my form.
Add a rule to the dropdown such that it executes when it is not blank. With the ID stored in the dropdown, you can use a filter within Set a field's value to then extract the DisplayName from your secondary data source and save it in the second field.
Good luck!