Hi,
That search was not visible, because the CSS file including the styles is cached by the site, and the server is pushing an old version of it – thus it does not contain the 3. search instance.
To bypass that issue, I have enabled the inline styles option on the plugin back-end here. Now the search box is properly displayed: https://i.imgur.com/gpQzvio.png
Once all the cache is properly cleared, you can turn off this option, and you should see the same layout as now. If it’s different, then the cache is not cleared. I highly recommend turning off all layers of cache until the site development is finished, it will conflict with any asset files, scripts/styles as well as page layouts. Please note, that caching is not something the plugin can control in any way. I am seeing a very aggressive cache on the front-end, even the media query strings are removed from the asset files. I am not sure if that is server side, or CDN, but in case you have that cache control on cloudflare/other CDN, I suggest temporarily disabling it until the development is finished.
I am honestly trying to help you out here, even though the problem is unrelated to the plugin itself. I know implementing/changing things on a website can be frustrating, and often times the cause of the problem is something else. If you turn off all the cache until you finish the changes on the site, you should see the changes immedately, and these kinds of issues will be less frequent.