add post type pram to wp shortcode

Do you want to be able to switch between post types?

If not, seems to me you could just duplicate the same functionality as you’re using with the per_page parameter, meaning:

In your shortcode function just add a data attr to the #container-async div for the post_type(s).

<div id="container-async" data-paged="<?php echo $a['per_page']; ?>" data-posttype="<?php echo $a['post_type']; ?>" class="sc-ajax-filter">

(you’ll need to add it to your shortcode_atts as well by adding ‘post_type’ => ”)

Then in your Javascript on(‘click’) function add it to $params:

$params    = {
    'page' : $page,
    'tax'  : $this.data('filter'),
    'term' : $this.data('term'),
    'qty'  : $this.closest('#container-async').data('paged'),
    'pt'   : $this.closest('#container-async').data('posttype')
};

Finally, in your php function where you gather your $args array for WP_Query, get the value and then add in the post_type:

$pt = $_POST['params']['pt'];

$args = array(
    'paged'          => $page,
    'post_type'      => $pt,
    'post_status'    => 'publish',
    'posts_per_page' => $qty,
    'tax_query'      => $tax_qry
);