I'm not an expert myself, but I think I can give you some pointers.
Since you have your laravel app in dog.com, basically all your requests will get redirected to dog.com/public/index.php. (This means, dog.com/docs will try to go to a route /docs in your laravel app).
I think you need to setup something in apache/nginix (whichever you're using) so that you inform the server that any request to dog.com/docs should go somewhere different.