Your question is a good one. We know from previous debriefings that revealing information from alternate timelines is generally not considered breaking the Temporal Prime Directive. Not only do we have the episode with Kes--where the information was even considered potentially useful--we also have incidents with Harry Kim and Worf, just to name a few. Even if we argue that he can't talk the part with Icheb and Naomi might not known to be from an alternate timeline, their role is relatively unimportant and can be left out. The rest is definitely an alternate tiimeline--Chakotay knows that none of that ever happened or ever can happen. So it makes no sense that the TPD would prevent him from revealing information.
But is it absolutely necessary that the Temporal Prime Directive is what prevents Chakotay from saying anything? Could not Chakotay have another reason for not telling the story to Janeway? Is there something about what happened that he might not want to let Janeway know about?
I propose that there is: the fact that Janeway had an opportunity to go back and fix her biggest mistake, stranding the two crews in the Delta Quadrant. But then we have the problem I stated before: why couldn't he just keep just that bit of information to himself? Well, he's already had experience trying to do so, and was eventually cajoled by past-Janeway into telling all. Thus it is better to keep the whole thing a secret. The Temporal Prime Directive is something that he's recently dealt with, works as an explanation for him knowing about a future event, and allows him to keep everything he wants to keep secret.
We also get a hint that Janeway may realize this, with her joking use of the concept at the end of the episode. It could easily be her way of letting him know that she knows he's choosing to keep something from her, hence why turnabout is fair play.