TYPO3 Core

Interesting. Is this gone in v11 when you enable 'Subrequest page errors' feature toggle? This may not work out if extensions are not sub request ready (which will be enforced in v12), but it could be a start.

The source of trouble is in PageContentErrorHandler, which delegates the "content-type" header only.

See: https://github.com/TYPO3/typo3/blob/fc51ccbf2bb8a8c959aa74cbceca124971e6e7fd/typo3/sysext/core/Classes/Error/PageErrorHandler/PageContentErrorHandler.php#L122

Would propose to delegate all headers from sub-request response.

There is the temporary solution:

<?php

/*
 * This file is part of the TYPO3 CMS project.
 *
 * It is free software; you can redistribute it and/or modify it under
 * the terms of the GNU General Public License, either version 2
 * of the License, or any later version.
 *
 * For the full copyright and license information, please read the
 * LICENSE.txt file that was distributed with this source code.
 *
 * The TYPO3 project - inspiring people to share!
 */

namespace ......\PageErrorHandler;

use Psr\Http\Message\ResponseInterface;
use Psr\Http\Message\ServerRequestInterface;
use TYPO3\CMS\Core\Error\PageErrorHandler\PageContentErrorHandler;

/**
 * Class PageNotFoundHandler, extends core functionality of {@link PageContentErrorHandler}.
 * It reuses the headers of response on sub-request.
 * 
 * NOTE: Can be removed if https://forge.typo3.org/issues/96294 is solved.
 */
class PageNotFoundHandler extends PageContentErrorHandler
{
    /**
     * @var ResponseInterface|null
     */
    protected ?ResponseInterface $responseOfSubRequest = null;

    public function handlePageError(ServerRequestInterface $request, string $message, array $reasons = []): ResponseInterface
    {
        $response = parent::handlePageError($request, $message, $reasons);
        if ($this->responseOfSubRequest instanceof ResponseInterface) {
            foreach ($this->responseOfSubRequest->getHeaders() as $headerName => $headerValue) {
                $response = $response->withHeader($headerName, $headerValue);
            }
        }
        return $response;
    }

    /**
     * @inheritDoc
     */
    protected function stashEnvironment(callable $fetcher): ResponseInterface
    {
        $this->responseOfSubRequest = parent::stashEnvironment($fetcher);
        return $this->responseOfSubRequest;
    }
}