fervent plinthYou can’t build a “new chat screen” inside SpiffyHUD because chat is part of the HUD, not one of the menu GUIs FancyMenu/Spiffy can re‑parent. What you can do is tweak how the chat overlay looks and where it renders by adding the Chat Customizer element to your HUD layout—position it, recolor it, add borders, etc. That’s the only custom chat control Spiffy exposes.
So the short version: you can’t replace the chat screen with a totally custom screen, but you can adjust the existing overlay via the Chat Customizer element.