We still use douzaines for eggs, oysters, snails and a few other delicacies but for “a moderate number of times”, while une douzaine is certainly idiomatic too, my impression is une dizaine prevails.

Google Ngram shows the metric system in action:

I understand you are looking for colloquialisms.
In this instance, “douzaine” is better if you want to express, let’s say, annoyance. But what would come naturally to me is “une demi-douzaine”
– j’ai dû lui répéter ça une demi-douzaine de fois
numerically inferior but rhythmically superior sentencewise, if you see what I mean.