I would be running the OS through an USB but I would have a portable SSD to host the blockchain and keep it updated there.
Theoretically, this can work. I haven't attempted to run Bitcoin Core within Tails, so it's possible that it requires specific configuration settings.
I would be accessing the CEX with clearnet. So when I send to a CEX, I would like to be able to send through clearnet, so it matches the clearnet IP with the broadcast IP. If I send it through my node, then it would go directly into the CEX node, so they would know the IP, or it goes through additional hops? I've read people got transactions blocked from broadcasting through Tor into a CEX.
I believe there's a misunderstanding. Broadcasting means sending to multiple nodes, and none of these nodes can determine if your node is the originator of the transaction. Even if your node is connected to a CEX's node, there's no method for them to identify you as the transaction owner.
Users are blocked when accessing CEX
from the Tor Browser, not when they broadcast transactions through the Bitcoin network using Tor.