JSON API not showing full content

You can try this to show the full post content only in the JSON API:

function remove_more_wpse_96740($content) {
    if(get_query_var('json')){
        global $post;
        $content = preg_replace('/<!--more(.*?)?-->/','',$post->post_content);
    }
    return $content;
}
add_filter('the_content', 'remove_more_wpse_96740',1);

by targeting the json query variable and remove the <!--more--> part of the post.

If you have a custom string like <!--more But wait, there's more! -->, it will also be removed.

This part of the plugin in /json-api/models/post.php:

  function set_content_value() {
    global $json_api;
    if ($json_api->include_value('content')) {
      $content = get_the_content($json_api->query->read_more);
      $content = apply_filters('the_content', $content);
      $content = str_replace(']]>', ']]>', $content);
      $this->content = $content;
    } else {
      unset($this->content);
    }
  }

is fetching the content and the check for <!--more(.*?)?--> in the content takes place in the WordPress function get_the_content().

PS: Using the method above will disable <!--noteaser--> in the JSON API, but that should be alright since you want the full post content there.