Chat suspensions are... strange. There's two ways to boot someone off chat - using a suspension on the parent account that's associated with the chat account (And otherwise it's not global)
Suspending someone only on chat has a few quirks on its own - we have no way of actually privately informing a user where they went wrong (and as someone who helps moderate a few other site chats, that's a pain), for some inexplicable reason we set suspensions by the hour. It's probably easier to set a site suspension if you're a moderator on the user's chat parent (or you reparent them), but misbehavior can be chat only.
While I realise chat isn't a priority right now, could we consider having moderator messages of some sort and by day, not per hour suspension options?