#You can use it. Once.
It's a spell, and it deals damage of the chosen type (assuming you choose the same type for the dragon's breath as you have for your bloodline). That meets the requirements for applying the class feature to that casting of the spell. That, as stated in the class feature, allows you to add your charisma modifier to one damage roll of the spell - in this case one exhalation. Any other breath attack would be different damage rolls, and wouldn't get the benefit. Timing here could get kind of odd, amusingly, as it never tells you you have to add it to the first damage roll of the spell, and it doesn't tell you if you decide whether or to add it before or after you roll the dice, but the "Yes it works, use it once." part is pretty clear.
It could actually be weaker than that. Your DM could decide, based on the text of the spell, that each target of each exhalation gets their own damage roll, in which case the bonus would only apply to one target of one exhalation. This DM wouldn't limit it to that degree, if only because thematically, Draconic Sorcerers really should be particularly good at using dragon's breath, but it would not be an unreasonable interpretation.
