Yes, just set the background-image url to the full path of the background image on the site that will be changing, then just make sure that any time you change the background image, you keep the filename the same and overwrite the previous background.
Set this code on the site that will be changing:
body {
background :url('/images/bg.jpg');
}
And this code on the site that won't be changing(change the URL to the absolute path of the background image from the dynamic site):
body {
background: url('http://yoursite.com/images/bg.jpg');
}