FileMaker does not use object names (field name, name of layout, etc.) to track objects in its various schema. Instead, it uses a unique, internally-generated ID. (There are some who argue that it should parse out the name, but that's a discussion for a different day.)
What you're seeing is an effect of that operation. By copying / pasting the new field into the data files, you're, in essence, assigning it a new unique ID in those files. The data file doesn't find the original ID to which it was pointed when it was created; hence, you get "<Missing Field>".
There are a few ways to handle this:
1) Perform your updates on a single copy of the solution, then duplicate that copy over for each client. (This would be a normal versioning process, and would likely involve a data migration.)
2) Create the new field independently in each copy, hence preserving the unique ID releationship between data and interface file.
3) Do what you're doing now - repoint the interface file to the new field.
Which option you select will largely depend on how complex the change is, how many copies you have, and how many instances of the field exist in the interface file.