Thank you all so much, these are wonderful options!
How would I do this, exactly?
From Chatty, you open a Stream Chat window (it's for display-only; additional to your primary IRC chat window). It's "Extra => Stream Chat => Open Dialog" from the menu.
Then from a scene in OBS MP, add a "Window Capture" type of resource. Select the right Window from the resource properties. It'll be something like "[javaw.exe]: Stream Chat" and show the chat in the preview window.
Now, you'll have that window in your scene and you can size it and adjust it however, like any other resource. You'll probably want to crop out the scroll bar. And if you look into the help docs for Chatty, you'll find some commands that allow you to configure the Stream Chat further (want messages to vanish after a few seconds or minutes? Want moderation messages to be included?).
Chatty also does some other stuff that I find useful, like letting you set/change your channel's game and stream title, see a list of new followers, subscribers, current (and historical) number of viewers, etc.
The only downside is... it's a java application. Still, it isn't too bad on resources.