Hi Bojan,
I do have a same problem and tried to fix it with your additional CSS. It works well when I change the header picture in a .jpg file, but doesn’t do it with an .png.
Do you have a solution? https://www.heleenvanhaaren.nl
This is what I added so far:
@media screen and (max-width: 600px){
div#site-header {
background: url(your_image_url);
}
#site-header img {
visibility: hidden;
}
}