Apache .htaccess rewritebase not working for subfolder

I am currently working with an old vanilla PHP codebase and I am supposed to make a few improvements.

I am supposed to create some new APIs and but unfortunately at the moment, the logistics of introducing a framework like Laravel or Symfony are not quite simple.

So to make things something work, I have created a folder called api under the root directory and added a .htaccess file under it with the following content

<IfModule mod_rewrite.c>
    <IfModule mod_negotiation.c>
        Options -MultiViews -Indexes
    </IfModule>

    RewriteEngine On
    RewriteBase /api/

    # Handle Authorization Header
    RewriteCond %{HTTP:Authorization} .
    RewriteRule .* - [E=HTTP_AUTHORIZATION:%{HTTP:Authorization}]

    RewriteCond %{REQUEST_FILENAME} !-d
    RewriteCond %{REQUEST_URI} (.+)/$
    RewriteRule ^ %1 [L,R=301]

    RewriteCond %{REQUEST_FILENAME} !-d
    RewriteCond %{REQUEST_FILENAME} !-f
    RewriteRule ^ index.php [L]
</IfModule>

and under it, I also have an index.php file, where I hope to put my endpoints with following code

<?php

use Dotenv\Dotenv;
use Psr\Http\Message\ServerRequestInterface;
use Laminas\Diactoros\ServerRequestFactory;
use Laminas\HttpHandlerRunner\Emitter\SapiEmitter;
use Laminas\Diactoros\ResponseFactory;
use League\Route\Router;
use League\Route\Strategy\JsonStrategy;

define('BASE_DIR', dirname(__DIR__));

require BASE_DIR . '/vendor/autoload.php';

$dotenv = Dotenv::createImmutable(BASE_DIR);
$dotenv->load();

$response_factory = new ResponseFactory();
$strategy = new JsonStrategy($response_factory);
/** @var Router $router */
$router = (new Router)->setStrategy($strategy);

$router->get('/', function (ServerRequestInterface $request): array {
    var_dump($request);

    return [
        'name' => 'kelly',
    ];
});

$request = ServerRequestFactory::fromGlobals(
    $_SERVER,
    $_GET,
    $_POST,
    $_COOKIE,
    $_FILES
);
$response = $router->dispatch($request);

// send the response to the browser
(new SapiEmitter)->emit($response);

I am using the thephpleague/route to handle my routing and a few other packages to handle other needs.

But when I visit http://backend-demo.localhost/api/ I get the following

backend-demo.localhost is configured as a virtual host in my local apache setup.

My guess is the 404 status is being caused by the .htaccess configuration.

What can I do to make this work?

Thank you.