Not sure what you're asking... here's a brief example/outline of what I was referring to:
EDIT: I'm an idiot. I forgot that the whole problem here was that you're including across HTTP. In order to do this, you'd actually have to use some output buffering (e.g. enable output buffering, include the remote file, grab & clear the output buffer's contents, str_replace() '{BROWSER_NAME}' with the correct data, and then re-output the correct data).
Note that another option would be to send the user-agent string to the included script via the query string, e.g.
include('http://mysite.com/include_me.php?agent=' . urlencode($_SERVER['HTTP_USER_AGENT']));
In the included script, you would then use the agent string passed via the query string as the first parameter for the [man]get_browser/man function.
EDIT2: Also note that since you're getting the data across HTTP, there's really no reason to use include() at all - a simple [man]file_get_contents/man or [man]readfile/man would suffice instead.