Why are you copying the name to another field? Sorry, don't understand what you're trying to do.

That should never be an issue. The Name field is always populated when first, middle, or last name fields are changed unless something is interrupting that process. I was not able to confirm any issue with this behavior. How are your users editing themselves? They should be exclusively using CBs profile edit.

So you're adding the new last name to a custom text field? If that's the case that explains your issue since it's a custom field it's not considered for name construction. You don't need a separate name field for this. You can use CB Conditional to show your document file field (make it required) if the lastname fields value changes. This is done using the below condition on your file field.

Display: Field conditional show
Field: Last Name (lastname)
Operator: Not Equal To
Value: [lastname]
Locations
Registration: No
Profile Edit: Yes
Profile View: No
Userlists Search: No
Userlists View: No

Now if they try to change lastname the file field should show up and since it's required they'll have to provide the documentation to change the lastname. This will keep all name fields properly in sync and won't require another text field.