I think the delay is due to Twitch itself.
Twitch uses RTMPE for the most part, it's always going to be a little delayed. It's a decent box, compression is on, connected to fibre internet.
If anyone has any ideas of reducing latency I'd love to hear it!
The delay experienced is all in the stream broadcast back to the Twitch servers and on to the viewers.
When I'm watching the chat and also watching the local view of Doom the commands are executed almost instantly. It then takes ~8s to show on my Twitch view. When I started it was around 15s but some tweaking in OBS has brought that down a bit.
Or you could execute the command for a second, then pause the game execution. Make it closer to turn-based Doom.
But as mentioned, it would have to be capped to some number to prevent abuse.
I am unfortunately completely ignorant of how you actually send data to Twitch, but can you use H264? It has an "ultra-low latency" mode.
Edit: Hmm, you're on OBS. Maybe you could stream using some uncompressed low-latency option from OBS to ffserver, then reencode on ffserver to low-latency H264. Alternatively, you could use ffmpeg with its X11 capture option, pointed at eg Xvfb.
And Discord just rolled out a beta for screen-sharing, also something to keep an eye on maybe.