Yes, but if you can affect what others know about you by actually ceasing to be "rational", and it will be profitable, persisting in being "rational" is harmful.

So it can be irrational to be rational, and rational to be irrational? Hmm. I think you might want to say, rather, that an element of unpredictability (ceasing to be predictable) would be called for in this situation, rather than "irrationality". Of course, that leads to suboptimality in some formal sense, but it wins.