Hoping not to sound too technical and seeing that you have already tampered with the CSS directives, my approach would be to restore the rules for the site header, only changing the CSS selector to target the header of one specific page instead of all of them.
The way to do this is simple, your theme is automatically adding a series of classes to the body tag on each of your pages. If you navigate to any of your pages and hit Ctrl+U (or Cmd+U on Mac) you’ll see the actual code of the page. Find the body tag (I suggest doing a Ctrl+F – Cmd+F for Mac – and searching for “<body” without the quotations). What you’ll see is that your body tag has a long string of classes. Find the “page-id-xxx” (where xxx is a unique number for that page) and use that as your unique selector in the CSS.
Your CSS should look something like:

.page-id-xxx .site-header {
  background: url(path to the background image);
  // more rules
}

Hope it helps