Sorry, after a few more trials and errors, I figured it out!
In random.php, I simply changed
Code:
# Get products data for current category and store it in an array
#
$cat = intval($cat);
to
Code:
# Get products data for current category and store it in an array
#
$cat = 8;
Where the number 8 above is the categoryid for the category who's products I want to display. Hope this helps someone!