If requests for the other domain name are simply redirected, there is no simple way to track the clicks through it. It doesn't serve a HTML page, therefore there is no content served on that domain that can be tracked.

You add only one tracking code to your page, the one for the final domain - the one that actually displays the content.

I don't know if this is feasible for your site, but if you can get the other domain to redirect to a specific single specially created landing page on the new domain, you could track visitors passing through that page instead. For this to work, the new page would NOT be linked from anywhere else on the main domain, but it would link to many pages of the new site: the major category pages at the very minimum. You would also need to "noindex" that page so that people do not find it via searchengines.