You can bind an exposed port to a specific IP by prefixing the IP address like this “-p xxx.xx.x.360:80:80”. Should work in a compose file “ports” list as well.
For outgoing traffic it’s a bit trickier, but if you create a separate network I think it should be possible to inject an iptables SNAT rule to use a specific source IP. Might be handy to make sure you’re sending emails from the correct IP but with your setup I would just make sure to use the primary IP as email egress…
This really has nothing to do with Smart TVs in itself though… It’s just a problem if you choose to play YouTube videos on your TV, which seems like a pretty reasonable thing to want to do.