Assuming I understand the problem correctly, you need to have the same column name in the output file twice. Doesn't matter whether it's same data or not, just the header needs to be repeated.
It's a little hokey, but in your connection manager, uncheck "Column Names in the first data row" and redefine the columns as email and email1. This will allow you to connect the columns to the right places in the file.
In your flat file destination, you have the ability to define Header row(s). It's very limited, you can't put useful things in there like dynamic checksums and such but in your case, paste in first1, last1, email, shortname, email
and run the package. Data will be extracted to the correct columns and a header row will be prepended to the file with all the "right" field names.
Two downsides to this approach. First is the connection manager becomes output only as it would attempt to read in the header row from the file. Second is that any changes to the layout will not be kept in sync with the manual header row.