Can I Hook Into the TinyMCE Insert/Edit Link Button to Use Shortlink For Post?

If you mean the link dialog, then we can modify the permalinks with the wp_link_query filter:

add_filter( 'wp_link_query', function( $results )
{
    foreach( $results as &$result )
        $result['permalink'] = wp_get_shortlink( $result['ID'] );

    return $results;
} );

where we use wp_get_shortlink() to get the short links.

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