I admit that I do not know the full workings of how the chatroom manages users from the same IP address, but it probably does block that to prevent nefarious behavior (for example, a troll logging in as two people at once and harassing other users).
The only thing I can suggest is if you can use a cellular phone as a hotspot that you connect one computer through it and then use your landline internet from the other. The phone, going through the wireless carrier's network rather than your landline, should have a different IP. Beyond that, I can offer no solution, I am afraid.