Get full content from feeds and save a duplicate copy in my site

You can do this on background using some ajax on page load, here is an untested example to show the idea.

jQuery(function() {
    jQuery.get('http://domain.tld/remote');
});



<?php
function is_ajax()
{
    return (isset($_SERVER['HTTP_X_REQUESTED_WITH']) and strtolower($_SERVER['HTTP_X_REQUESTED_WITH']) == 'xmlhttprequest');
}

function add_rewrite_rules($wp_rewrite)
{
    $new_rules = array
    (
        'remote' => 'index.php?remote=true'
    );

    $wp_rewrite->rules = $new_rules + $wp_rewrite->rules;
}

function add_query_vars($vars)
{
    array_push($vars, 'remote');
    return $vars;
}

function remote_feed()
{
    if (($var = get_query_var('remote')) and is_ajax())
    {
        // get remote feed and insert/update posts
    }
}

add_action('generate_rewrite_rules', 'add_rewrite_rules');
add_filter('query_vars','add_query_vars');
add_action('template_redirect', 'remote_feed');