Loop in elementor custom widget not working

I would recommend switching to a standard post loop:

    $args  = [
        'posts_per_page'  => 12,
        'category_name'   => 'featured',
        'orderby'         => 'post_date',
        'order'           => 'ASC',
    ]; 
    $query = new \WP_Query($args);
    if ( $query->have_posts() ) {
        while ( $query->have_posts() ) {
            $query->the_post();
            .....
        }
        \wp_reset_postdata();
    }

Note that I:

• switched to modern PHP array syntax of [] rather than array()
• removed the suppress filters parameter
• switched to a standard \WP_Query loop
• checked if any posts were actually found, providing a way to add an else statement with a “no posts found” message
• switched to standard PHP syntax with {} brackets rather than the alternate syntax
• fixed the indenting
• called the_post() to set the current post, equivalent to setup_postdata( $post ); but cleaner
• called wp_reset_postdata() to clean up after the loop so that the next piece of code gets the correct post, not the last post the loop iterated over
• removed the post_type parameter as the default post type is already post
• You mentioned in a comment there was a namespace related error, so I added \ to ensure it pulled WP functions from the global namespace

I do have 2 unrelated notes though:

Return Values and the_ functions

I spotted these:

            echo the_post_thumbnail('large');

and

            echo the_title();

These functions don’t return a value to echo, that’s not how they work. They are equivalent to:

echo '';
the_post_thumbnail('large');
echo '';
the_title();

So don’t use echo, the same goes for any other WP function beginning with the_ such as the_content, etc. If you really need the value returned so you can put it in a variable, use the equivalent get_ style functions.

Carousel Markup

If I understand right, if 5 posts are found, there will be 5 carousels, each with a single post in them. I would move the carousel container markup to before and after the loop, so that inside it you’re only outputting the current slide/item.

In general though, carousels have apalling UX, poor accessibility, high complexity, and a high overhead on performance and CSS/JS weight. You would be better off with a hero pattern, or a featured grid of some sort, carousels have extremely poor discoverability, so any carousel will be just as effective as showing the first item and not showing the rest. Setting 'posts_per_page' => 1 will be just as effective.