Hi,
There is sort of a “hacky” way, by configuring the search on the site where the post types exist ( https://bernstage.com/ ). You should be able to access the search options on both backends. Select the post types on the back-end where they are available. Once you configure it, don’t make changes to it on the other site. The plugin should still memorize the non-existent post types, and should be able to access them from the other site as well. Let me know if that does anything. If not, then I could maybe suggest a custom code, but I can not guarantee it will work.
Optimally it’s better if all post types are registered on both sites to make sure the plugin can cross-check with the WordPress core.