Hi,
I don't know all the features of chameleon. Maybe they have something in place to be able to do that easily.
So I would recommend to first check with them.
Otherwise, with a small system plugin, you could just get the partner_id parameter in the URL and store it $_SESSION and then, on each page, you would then get it from $_SESSION and set it back to the URL in $_GET and or $_REQUEST and then chameleon should be able to use it for its filters (to confirm with them). In that case, it would only require something like 10 lines of code in the plugin.