It should also be noted that the message limit is per connection, not per IP, so one workaround could be to open multiple concurrent connections and split the messages to be sent between them.
Even if you do this though you should still implement a message queuing system that either drops messages entirely when one connection is nearing the limit (as the ban is account based, so if one connection breaks it they all get banned) or delays messages being sent until there has been sufficient time since previous messages. The number of connections should also be limited so that large channels (or channels with malicious users trying to force the bot to spam) don’t get too excessive.
I don’t know what sort of game requires that many messages that you’re frequently running into this being an issue, but other consideration might be to rethink what messages you send to the channel, for example are all the messages essential or could some be skipped if it’s a duplicate of another recent message? could the streamer potentially have options to enable/disable certain message types?