If you'd like to participate in the chat but want to change your user name, you can do so from the the "Settings" menu. To access the "Settings" page, click on the circle with three lines on the top left of the bitFlyer Lightning page.
Click on "Settings" and scroll down to the "Chat Settings" section. From there, you can create a new nickname.
Try to create a unique nickname as common ones may be unavailable. When you've finished creating your new username, you can go back to the Trade page and your name will be saved.