‘Trick’ a plugin (WP-Members) to think the blog language has changed for a single page

Yes, sort of. Any localized plugin such as WP-Members runs off of the “locale” that WordPress is set to. This value can be filtered for WP as a whole using the locale filter. But it can also be filtered for plugins using plugin_locale

See: https://developer.wordpress.org/reference/hooks/plugin_locale/

To use plugin_locale specific to the text domain for the plugin* (which in this case is ‘wp-members’)

add_filter( 'plugin_locale', 'my_plugin_locale_filter', 10, 2 );
function my_plugin_locale_filter( $locale, $domain ) {

    // If the text domain is 'wp-members'
    if ( 'wp-members' == $domain ) {
        /*
         * This logic adapted from the method you referenced at
         * http://beta.beantin.se/wordpress-setting-language-individual-pages/
         * Note that the function get_top_parent_page_id() from that example
         * is used here, and the logic is not adapted to your specific 
         * question (i.e. I assume the default $postLanguage value is
         * Norwegian in your case and you are switching it to English) so 
         * change this to match what you are doing with the other elements
         * being used from that process.
         */
        $postLanguage = "en-GB";
        if (is_page()) {
                $svPageID = get_top_parent_page_id(); // ID of parent page
                if ($svPageID == "565") { // ID of the "på svenska" page
                    $postLanguage = "sv";    
                }
            $locale = $postLanguage;
        }
    }

    // Always return the value being filtered.
    return $locale;
}

*The text domain for any plugin should be noted in the header of the plugin’s main file. For example, if you open wp-members.php and look at the plugin header, you will find this listed under the heading “Text Domain:”.