The answer is no unless there is some undocumented headless interface for Google Chrome extensions.
But if you got SSH running you can install a VNC server on the target, and then tunnel a VNC connection through SSH and get a visual desktop this way. You'll need to forward port 5901 (by default you will be running on the first virtual desktop) and then point your vnc client (e.g. tightvnc) to 127.0.0.1:1 (or 127.0.0.1:0 if you'll be using the real desktop via x11vnc).
Then you will be able to set a PIN in Chrome Remote Desktop visually. And then laugh at the whole setup of having to use VNC to connect via Chrome RD, but this is how I tested that the Chrome client actually works.