Hi,
Well, the only way is to add the same shortcodes to the 404 page template. If you have a theme builder of some sort, then it may allow edigint the 404 page template. Otherwise the 404 template file needs to be edited directly and the search form replaced with the PHP version of the shortcodes. I would only recommend doing that, if you are comfortable with editing files.
If you want, you can add temporary FTP access to your ticket, and I can try replacing that for you.