Hi Will,
The thing we are talking about is commonly called “domain parking”. But you also have to instruct TCH to accept queries for your .org domain. If you host website.com at TCH (or somewhere else) and then change the nameservers for website.org to point to TCH nameservers, they will not respond to DNS queries for your .org domain. Even more, the web server will not know that your .org site is the same as .com one. So before rushing to make the change at your registrar, be sure to get all the information from TCH support.
Kind regards,
just X