I tried just using the bot account on twitch and posting a message to the chat where the messages were being dropped. I could post messages normally and got no warning or notification. However, no one could actually see the messages, including myself on my normal account logged in with a different browser.
So sending too quickly or duplicates is not the problem. I have looked at the raw IRC logs and there’s nothing to indicate that messages aren’t going through.
The problem seems to only affect the bot account, because I can post messages with my normal account with no problem (same IP address).