it does become a data migration exercise as you are physically moving data, but since the schema is the same there should be no issue.
You will need to drop FKs and recreate them etc, but if you use a schema comparison tool post implementation (e.g. redgate data compare) this should alleviate any business concern.
The way I see it, they dont have any choice because you will run out of space soon and everything will stop working.
The only other option is that you archive data, e.g. if your DB contains historic data which is not required through your application but you need to store in a DB, you can generate an exact schema replica (e.g. where you have table TABLEA on primary filegroup on your first data file, create a new table ARC_TABLEA on the new filegroup on the second partition file. Move all data which meets your archive rules.)
There are many things you could do, but ultimately splitting the data is going to need to happen if you cant increase the disk size. (Its also much better for performance )
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