What you're looking for is commonly referred to as a 'dump button'. It's a requested feature, but due to how OBS Studio is coded, and how streaming works, apparently is not feasible to add at this time.
The only real option at present is what Tomasz listed; add a delay in Advanced, and stop the stream (with the setting to stop IMMEDIATELY, otherwise it will play out the delay buffer before stopping). Not great, but if you ABSOLUTELY CANNOT SHOW some event on-stream, it's a handy emergency ripcord.