I need posts within a taxonomy category that are tagged “featured” to show up first

I really hope somebody can give you a much simpler answer, but this is pretty much all I can think of at the moment.

If you’re okay with all the listings being on a single page you could do something like this:

<?php
// Get the slug of the page to use in our category argument 
global $post;
$post_slug = $post->post_name;
?>

<?php
$args = array(
    'category_name' => $post_slug,
    'post_type' => 'businesslistings',
    'posts_per_page' => -1,
    'orderby' => rand,
    'tag' => 'featured'
);
$featured_query = new WP_Query( $args );
?>

<?php if ( $featured_query->have_posts() ) :  ?>
<?php while ( $featured_query->have_posts() ) : $featured_query->the_post(); ?>

    // Display featured stuff

<?php endwhile;?>
<?php endif; ?>

<?php
$args = array(
    'category_name' => $post_slug,
    'post_type' => 'businesslistings',
    'posts_per_page' => -1,
    'orderby' => rand,
    // x below should be replaced with the featured tag's ID (link for instructions will be below)
    'tag__not_in' => 'x' 
);
$standard_query = new WP_Query( $args );
?>

<?php if ( $standard_query->have_posts() ) :  ?>
<?php while ( $standard_query->have_posts() ) : $standard_query->the_post(); ?>

    // Display standard stuff stuff

<?php endwhile;?>
<?php endif; ?>

References:

Using WordPress tags in the WP_Query: WordPress Codex

Getting the page slug: Stack Exchange

Getting your tag’s ID: Stack Exchange

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