Maybe something as simple as a textual overlay "Don't worry - the stream will be back soon" and a script to internally hammer the only element of the page that's actually needed for a video refresh
This could be achieved if twitch allowed broadcasters to upload a 2 second loop as a 'placeholder' for connection drops, and twitch could mix that in if it detects too many frame-drops.
It would need to be a setting though because some streamers just stream over a bad connection (e.g. Hitchhiking streams) and wouldn't want that interruption.