Is it possible to have such a visible column in SQL Server filled with automatic values, same as the hidden uniquifier column?
No there is no built-in feature to do this.
Or do I have to manually calculate such a column when I insert rows?
You could consider using INSTEAD OF or AFTER triggers to maintain the correct uniqueifier-like behaviour for the same-time group when rows are inserted/updated/deleted.
This may require some careful thought to operate correctly under high concurrency, but it is probably best done as close to the data as possible (i.e. server-side, in the database, rather than client-side). A well-written trigger (with supportive indexing) will introduce minimal overhead.
Alternatively, make the index unique and implement retry logic. Since the expectations seems to be that collisions will be rare, this might well be acceptable.