The Dragonborn's breath weapon has a built in limitation:
After you use your breath weapon, you can’t use it again until you
complete a short or long rest.
Unless the dragonborn is taking a short rest after every combat and you, the DM, are allowing this short rest to go uninterrupted, the Dragonborn only gets to use this a few times a day.
I would expect that the Drow are not permitting the escapees much time for resting during and shortly after their escape.
If you're concerned about the dragonborn using his power to break free, well, the breath weapon only affects creatures:
When you use your breath weapon, each creature in the area of the
exhalation must make a saving throw, the type of which is determined
by your draconic ancestry...A creature takes
2d6 damage on a failed save, and half as much damage on a successful
Furthermore, if the dragonborn is planning to escape with the rest of the group, why would he use his breath weapon to break free of the door or his manacles when he could potentially use it to the group's benefit once fighting breaks between the captives and the guards? Seems like the more strategic use would be in the fighting during the escape, not to break out of the pen itself.
In any case, as far I see it, there's no real concern here -- the characters are expected to escape, after all.