The image CSS is also used in Property full width slider. If you modify that CSS, and you’ll affect the slider.
In the example you showed me, the image height is large).
I recommend to do the same.
Add an image with bigger height in header.
Add a second mobile CSS position for the advanced search on ipad resolutions.
For smaller resolutions, you’ll be able to use the Advanced Search shortcode – help here http://help.wpestatetheme.org/#space
This shortcode always shows on page (web and mobile).
If you give me access to your test domain, I can show you an example on a test page. It is the easiest way, as I mentioned above.
Hope this helps 🙂