Creating Non-Clustered Index on Non-Persisted Computed Column SQL Server
When SQL Server creates the index on the computed field, the computed field is written to disk at that time - but only on the 8K pages of that index. SQL Server can compute the InvoiceStatusID as it reads through the clustered index - there's no need to write that data to the clustered index.

As you delete/update/insert rows in dbo.Invoice, the data in the indexes is kept up to date. (When InvoiceStatus changes, SQL Server knows to also update IX_Invoice.)

The best way you can see this for yourself is to actually do it: create these objects, and execute updates that touch the InvoiceStatusID field. Post the execution plan (PasteThePlan.com is helpful for this) if you want help seeing where the index updates are happening.