I don’t see anywhere in their API where they provide a size larger than 640: https://instagram.com/developer/
1080px images are not yet officially supported via the instagram API, however you can access them by replacing the size in the image url.
A quick hack for this plugin:
wp-content/plugins/dsgnwrks-instagram-importer/dsgnwrks-instagram-importer.php
around line 1116 you see this:
$tmp = download_url( $media_url );
right before this line, insert
$media_url = str_replace( 's640x640', 's1080x1080', $media_url);
but be aware that this change will be reverted the next time you update the plugin.
I’ve updated the plugin to actually attempt to download the 1080 version and fallback to the standard resolution if it can’t find it. Thanks for the hint maxhaesslein.