You could try to modify the feed query directly via:
/**
* Modify the main feed query to show posts from the previous week,
* when the GET parameter wpse_previous_week is set
*
* @link http://wordpress.stackexchange.com/a/194843/26350
*/
add_action( 'pre_get_posts', function( $q )
{
if( $q->is_feed()
&& filter_input( INPUT_GET, 'wpse_previous_week', FILTER_SANITIZE_NUMBER_INT )
)
$q->set( 'date_query',
[
[
'before' => 'Sunday previous week',
'after' => 'Monday previous week',
'inclusive' => true
]
]
);
} );
and view the feed with:
example.tld/feed/?wpse_previous_week=1
Note that you might have to test and modify the date strings to better suit your needs.