Can I set the post_type query_var as a link is clicked?

You can use pre_get_posts filter hook to set the post type passed by the user:

paste this code in your theme’s functions.php file:

function user_set_type( $query ) {
    //only on your taxonomy page
    if ( $query->is_tax('YOUR_CUSTOM_TAXONOMY') ) {

        //and only if the guesst has selected a type
        if (isset($_GET['UTYPE']) && !empty($_GET['UTYPE'])){
            $query->set( 'post_type', $_GET['UTYPE'] );
        }
    }
    return $query;
}
add_filter( 'pre_get_posts', 'user_set_type' );

Change YOUR_CUSTOM_TAXONOMY to the name of your taxonomy, then on your taxonomy page or widgets all you need to do is create links with the post_type as parameters in them eg:

<a href="https://wordpress.stackexchange.com/questions/19524/<?php echo get_permalink() ."?UTYPE=designer"; ?>">Designers</a> - <a href="https://wordpress.stackexchange.com/questions/19524/<?php echo get_permalink() ."?UTYPE=boutique" ?>">Boutiques</a>