Search query -> Show all pages except certain template

REWORKED ANSWER TO ANSWER THE QUESTION

In order to exclude all pages with a certain template, all you need to do is to run a meta_query to exclude all pages with the custom field _wp_page_template set to bedankt. Remember, WordPress saves the template assigned to a page as a hidden custom field called _wp_page_template

With this in mind, we can do the following

$args = [
    's'         => $s,
    'post_type' => ['post', 'page', 'p24_cases'],
    'meta_key'  => [
        [
            'key'     => '_wp_page_template',
            'value'   => 'bedankt.php',
            'compare' => 'NOT IN'
        ]
    ]
];

ORIGINAL ANSWER – misread question

You simply need to get the post ID of that page, anf then pass it as an array to post__not_in

'post__not_in' => [1], // Replace 1 with actual page ID

Or pre PHP 5.4

'post__not_in' => array( 1 ), // Replace 1 with actual page ID