I would want to argue the central fact that supports the claim that Chanticleer (or Chauntecleer depending on the translation) is a round character is actually the way that he clearly shows he has learnt from his experiences with the fox by the end of the tale and also the way that he admits his own foolishness in giving in to the fox's flattery in the first place.
Let us just remind ourselves that a round character is a character who is shown to us in their full psychological complexity, rather than the rather two-dimensional nature of flat characters, who have but one or two character traits. We certainly see the full complexity of Chanticleer at the end of the tale when he talks to the fox when the fox asks him to come down:
You'll not, with your soft soap and flatteries,
Get me to sing again, and close my eyes!
To him who shuts his eyes when he should look,
And that on purpose, the Lord send bad luck!
Here we see Chanticleer being open about the way that he was almost fatally susceptible to "soft soap and flatteries," admitting his faults, but then also learning from this and integrating this into his character, showing that he is a round character rather than a flat one.