SQL Server FullText search on non-persisted columns
There isn't any kind of drawback with having a full-text index on a non-persisted column. Creating an index and putting a non-persisted column, the column will be automatically persisted on disk. You can try by yourself creating a nonclustered index and putting a non-persisted column. After this, you can use the command DBCC PAGE for looking the entire data of the nonclustered index and you will see the non-persisted column there.

Have one thing on mind. the non-persisted column stills non-persisted on the clustered-index.