I had an idea for a hat or a medal.
Chat Stalker Get up to 10k unread messages in the chat
Could be quite funny. And hard to do.
It is unlikely new hats will be added within the timespan of this event. (Though is there something else hat-based in our future? Time will tell)
Nor does the criteria you present seem any bit badgeworthy. Badges should be meaningful, not something that can be farmed by leaving your browser in the chat undisturbed for a week.