No, Breya’s ability cannot target Haktos if a number other than 4 was chosen.
Rule 702.16b says
A permanent or player with protection can’t be targeted by spells with the stated quality and can’t be targeted by abilities from a source with the stated quality.
In the stated situation, Haktos has protection from each mana value other than 3, and Breya has the mana value 4, so Breya’s ability cannot target Haktos.