What would I need to write into a custom plugin in order to add a switch for a custom string of CSS to the edit page?

This code should do the trick. Put this in your functions.php file.

function add_custom_css_meta_box()
{

    add_meta_box(
        'custom-css',
        'Custom CSS',
        'post_meta_checkbox',
        'page',
        'side'
    );
}
add_action('add_meta_boxes', 'add_custom_css_meta_box');

function post_meta_checkbox($post)
{
    $checkbox  = get_post_meta($post->ID, 'custom-css', true);
    $checked    = false;

    if($checkbox == 'true')
    {
        $checked = true;
    }

    ?>

    <p>
        <label>Tick this for custom CSS</label><br />
        <input type="checkbox" name="custom-css" id="custom-css" value="true" <?php if($checked == true) { echo 'checked'; } ?>/>
    </p>

    <?php
}

function save_post_meta()
{
    update_post_meta(get_the_ID(), 'custom-css', $_POST['custom-css']);
}
add_action('save_post', 'save_post_meta');

When you want to check if you need to load the CSS do an if statement and use get_post_meta($post->ID, 'custom-css', true) and check if it is true or false. Replace the $post->ID with how ever you’re getting the page id. You could do this on the page.php file.