Hi Michal,   

Currently it's not possible to do that with the Exporter.
The workaround is to use the Toolbox addon: Toolbox Addon

You can create a Manual Processing with those settings: 

Then in your Search Form, add a Submit button which is calling your Process:


You can also push some fields to the process and get them with a JFactory::getApplication()->input->get( '..' );

You can use this base for your process :
We are using the JCckContent class to get all items and extend this class with some specifics functions.
This exemple is working a Free Object.
Feel free to modify it if your are working on another object.

if ( !isset( $options ) ) {
    return false;
}

// Init
$app            =    JFactory::getApplication();
$format         =    $app->input->getString( 'my_field_name', 'default_string' );
$content_free   =    new JCckContentFree;
$path           =    JPATH_SITE.'/tmp/'.$config['uniqid'].'.csv';
$rows           =    array();

$content_free->setTable( '#__cck_store_form_my_free_table' );
$content_free->extend( __DIR__.'/extend/mixin.php' );

if ( $content_free->load( $config['pk'] )->isSuccessful() ) {
    $rows[]        =    $content_free->_getValues();
}
if ( JCckDevProcessing::isFirstItem( $config ) ) {
    if ( count( $rows ) ) {
        $header        =    array(
                            'id',
                            'title',
                            'date',
                            ....
                        );
        
        $content_free->_writeCsv( $path, array( 0=>$header ) );
    }
}
if ( count( $rows ) ) {
    $content_free->_writeCsv( $path, $rows );
}
if ( JCckDevProcessing::isLastItem( $config ) ) {
    $config['path']    =    $content_free->_getPath( $path, $format ); // The path of the CSV file
}

Here the base of a Mixin:

$mixin    =    new class() {
    use JCckContentTraitMixin;
    // _getPath
    protected function _getPath()
    {
        return function( $path, $format = 'csv' ) {
            if ( !JFile::exists( $path ) ) {
                return;
            }
            $info        =    pathinfo( $path );
            $new_path    =    $info['dirname'].'/my-export-'.JFactory::getDate()->format( 'Y-m-d' );
            $new_path    .=    '.csv';
            JFile::move( $path, $new_path );
            return $new_path;
        };
    }
    // _getValues
    protected function _getValues()
    {
        return function() {
            // Customize your query here
            $data    =    $this->getData();            
            $result    =    array();
            foreach ( $data as $key => $value ) {
                $result['o_'.$key]    =    $this->_get( $key, $value );
            }
            return $result;
        };
    }
    // _writeCsv
    protected function _writeCsv()
    {
        return function( $file, $rows ) {
            $handle    =    fopen( $file, 'a+' );
            foreach ( $rows as $row ) {
                fputcsv( $handle, $row, ';' );
            }
            
            fclose( $handle );
            return true;
        };
    }    
};

Hopping it will help.

Regards