Although the option to give up and be auto-hit is not explicitly denied, power descriptions are written assuming that threats are real and that all combatants are using their best efforts to hit and avoid being hit. For instance, the allocation of so many hit points of damage, as opposed to just dying when struck with lethal force, is because even when "hit" it is assumed you have deflected or avoided most of the blow, and are only truly vulnerable when all the energy to fight has gone from you.
Some powers work less well, or can be abused, if these assumptions are not played out.
The equivalent rule in the positive sense - that to gain the benefits associated with hitting an enemy, it has to be a real attack - is called the "Bag of Rats" rule.
There is no RAW for a character directly allowing themselves to be struck by an ally in order to gain a side benefit. In fact the assumption in the game rules is that even when they have been "hit" that they have managed to prevent themselves being outright killed by actively resisting or avoiding the attack.
With that interpretation, a particularly harsh ruling might say that character that simply gave up and allowed themselves to be struck by a lethal attack such as their ally's breath weapon should be killed outright (and using the Atonement should have a similar effect on the Dragonborn). The thing that prevents PCs being similar to minions in this respect is their pool of hit points that represents the character resisting this end result.
Just to be clear, I am not advocating that ruling, just pointing out that it would be logically consistent with game design.
I think that a character/team build that relies on this effect needs discussion with a DM before using it in play, as it is subject to DM interpretation of the rules. wax eagle suggested in comments that it might be reasonable to have the Dragonborn's ally expend an action to grant Combat Advantage to them, in order to set up the self-recharge.
Where you end up on scale from "fun tactic" to "over-optimised weirdness" I cannot judge - hopefully you have a DM who's willing to experiment and you can give it a try and figure out something that's fun for the group.