How to add a timestamp to the exported WXR filename?

I don’t think you need to do that, it’s already there (if I understand the question correctly):

In the WXR file we get a timestamp like:

that’s generated with:
```
<?php the_generator( 'export' ); ?>
```
that calls get_the_generator() where the export case is:
```
$gen = '';
```
It’s possible to modify it or append to it via the get_the_generator_export filter.

Additionally the timestamp is added to the xml document with:

<channel>
    ...
    <pubDate><?php echo date( 'D, d M Y H:i:s +0000' ); ?></pubDate>

The generated export filename also contains a datestamp:

$wp_filename = $sitename . 'wordpress.' . $date . '.xml';

Update

To modify the exported filename, one can use this filter:

/**
 * Filters the export filename.
 *
 * @since 4.4.0
 *
 * @param string $wp_filename The name of the file for download.
 * @param string $sitename    The site name.
 * @param string $date        Today's date, formatted.
 */
 $filename = apply_filters( 'export_wp_filename', $wp_filename, $sitename, $date );

Example:

Here we add the \TH-i-s part to the filename:

add_filter( 'export_wp_filename', function( $wp_filename, $sitename, $date )
{
    return sprintf( 
        '%swordpress.%s.xml',
        $sitename, 
        date( 'Y-m-d\TH-i-s' )
    );

}, 10, 3 );

This will generate a filename like:

sitename.wordpress.2016-10-22T19-26-07.xml

This will avoid a possible confusing download series like this one:

sitename.wordpress.2016-10-22.xml
sitename.wordpress.2016-10-22 (1).xml
sitename.wordpress.2016-10-22 (2).xml
sitename.wordpress.2016-10-22 (3).xml
...

Hope you can adjust this to your needs!

Update

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