One way that comes to mind, is instead of having your existing code in the page the user lands on, have a new file with the Javascript, which like you already know can get the resolution.
Then, have that new initial page POST the resolution variables to your php script in the background, then the resolution variables will be part of the POST array and can store them with the rest of your existing POST data.
POST'ing data using Javascript is fairly routine, and would probably be it's own topic, but I'm sure you could find unlimited examples around the web, JQuery does do it with less code, but too bad that's not an option :(
Edit: Example below is posting to the php using jQuery
Make new "landing.php" (doesn't have to be .php, could be .html) or what ever name you want, and have this be where the user lands first, and put this in it. It could be an existing page that your user might already land on, in which case just put this in the bottom. Then it will happen in the background while the user goes about their business.
<script type="text/javascript">
var screenWidth = window.screen.width,
screenHeight = window.screen.height;
$.post('name_and_path_of_php_file_you_already_created.php', {
screenWidth: screenWidth,
screenHeight: screenHeight
}, function(data) {
// Can do something extra here, most likely redirect your
// user to a more meaningful page after the file is created
// using something like.
window.location.href = 'some_meaning_page.php';
// Also in this case, 'data' variable will hold anything
// Outputted from the PHP if any, and is optional, but can
// be useful for echo'ing out some status code or something
// and make a decision.
});
</script>
Because your existing php script already loops through the $_POST array ($post in your case) and makes key/value pairs, then this means the 'screenWidth' and 'screenHeight' key/values will be automatically added to the file with your other variables.
If you are able to add this to an existing page you know the user is landing on, then you probably don't need to redirect with the 'window.location.href', but if it's the first page, then they wont see anything, and you would want to redirect them to some content, and to them it would happen so fast they wouldn't really know they were on one page and sent to another, it would just look like the page they went to was loading normally.
Let me know if this is not clear, or if need help with another aspect.