admin_notices action doesn't trigger within save_post action

You need to send your notice message – or probably a key that refers to a specific standard notice message – over the URL to the next page.

I’d advise encapsulating this in its own function. Something like…

public function render_to_pdf() {
    // ... whatever you need to do
      my_trigger_notice( 1 ); // 1 here would be a key that refers to a particular message, defined elsewhere (and not shown here)
    }

Then add your new function in functions.php (or an admin.php file):

function my_trigger_notice( $key = '' ) {
    add_filter(
        'redirect_post_location',
        function ( $location ) use ( $key ) {
            $key = sanitize_text_field( $key );

            return add_query_arg( array( 'notice_key' => rawurlencode( sanitize_key( $key ) ) ), $location );
        }
    );
}

Now when the page redirects, it should append your notice key on the URL, enabling the next page to “catch it” on the admin_notice hook (also set in functions.php or admin.php):

function my_admin_notices() {
    if ( ! isset( $_GET['notice_key'] ) ) {
        return;
    }
    $notice_key  = wp_unslash( sanitize_text_field( $_GET['notice_key'] ) );
    $all_notices = [
        1 => 'some notice',
        2 => 'some other notice',
    ]
    if ( empty( $all_notices[ $notice_key ] ) ) {
        return;
    }
        ?>
    <div class="error">
        <p>
            <?php echo esc_html( $all_notices[ $notice_key ] ); ?>
        </p>
    </div>
    <?php
    }

add_action( 'admin_notices', 'my_admin_notices' );

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