As I understand it, a copied file is considered to be a new entity, and therefore its owner will be the user or process that created it. Interestingly, a moved file will retain all of its attributes.
If moving (rather than copying) is an option, that should work for you. If not, you'll have to find a way to change the attributes on the newly copied file. Some copy utilities (such as Robocopy) have switches that allow you to retain some or all of the attributes. I'm not that familiar with SSIS, but maybe there's a way to call a batch file rather than use the native copy command?