wp-load Without Loading the plugins

You would need to put something like this in a .php file of any name in your must-use plugins folder (/wp-content/mu-plugins/) so that it loads before all plugins so you can use it to filter the plugin loading…

<?php 
if (defined('SHORTINIT') && SHORTINIT) {
    add_filter('option_active_plugins', 'shortinit_plugins_filter');
    function shortinit_plugins_filter($plugins) {
        $noloadplugins = array('w3-total-cache');
        foreach ($plugins as $i => $plugin) {
             if (in_array($plugin, $noloadplugins)) {unset($plugins[$i]);}
        }
        return $plugins;
    }
}

Alternatively manually include whatever function files you need to make things work as they should.

UPDATE for Multisite

You would need to use a different filter on multisite, something like this should work, eg. for blog ID 2…

<?php 
if (defined('SHORTINIT') && SHORTINIT) {
    add_filter('site_option_active_plugins', 'shortinit_site_plugins_filter', 10, 3);
    function shortinit_site_plugins_filter($plugins, $option, $network_id) {
        if ($network_id == 2) {
            $noloadplugins = array('w3-total-cache');
            foreach ($plugins as $i => $plugin) {
                 if (in_array($plugin, $noloadplugins)) {unset($plugins[$i]);}
            }
        }
        return $plugins;
    }
}

Note if wanted to change sitewide activated plugins you would need to do further filtering of the network wide active plugins, something like:

<?php 
if (defined('SHORTINIT') && SHORTINIT) {
    add_filter('option_active_sitewide_plugins', 'shortinit_network_plugins_filter');
    function shortinit_network_plugins_filter($plugins) {
        if (get_current_blog() == 2) {
            $noloadplugins = array('w3-total-cache');
            foreach ($plugins as $i => $plugin) {
                 if (in_array($plugin, $noloadplugins)) {unset($plugins[$i]);}
            }
        }
        return $plugins;
    }
}