Maybe show it when you press T button (if you press arrow in bottom left corner it would hide) or add another hotkey to show/hide it
Also i think it would be cool if you could export it with final video. Maybe alongside with sidebar (you can see it when you edit replay), crosshair and hotbar (if clients get information about other players' inventories of course)
I've added an option in the settings menu which will allow you to toggle whether chat messages are replayed with the next release (1.0.4). They won't be shown when exporting a video though as GUI exporting isn't within the scope of this project for now.