How to have different header.php based on the user logged in?

You can achieve this by getting the Membership Level of the current user, and making a conditional based on this. As I do not have a license for wishlist member, this is untested.

I suppose $member_id to be the same value as the WordPress ID of the logged in User.

<?php 
    global $current_user;
    get_currentuserinfo();
    $member_id = $current_user->ID;
    include('wlmapiclass.php');
    $api = new wlmapiclass('http://yourblog.com/', 'yourAPIKey');
    $api->return_format="php"; // <- value can also be xml or json
    $membership = $api->get( /members/{$member_id} );
    $membership = unserialize( $membership );
    print_r( $membership ); // see in which value the actual ID or slug of the membership is stored, I cannot reproduce that..
?>

As TomJNowell pointed out, it would be even better to change the include to

get_header( $membershiplevel );

So you are good to go for future Membership levels.