Why, precisely, is it irrational to respond to this?

Being angry signals lots of things - and if I desire less angry reaction, I need to figure out the function of the anger in this particular context.

Dahlen's points seems to be that in the ordinary social context, anger tends to function as an attention seeking behavior, not a conflict resolution behavior. In other words, most anger is trying to yank someone's chain. If that is the case, then responding to the anger with more anger is not consistent with having a goal of reducing the amount of anger directed at oneself.

Your assertion that anger reactions can have only one function seems likely to be false - and not a charitable or steelman reading of Dahlen's post.