I think this is balanced
Using the example in your comment:
So like if you are in a dungeon with hallways, you could burn straight down the hallway or if you are in a room, you could toast the room.
Since you can only use it once per short rest anyway, you can't do both without resting in between. Each case taken in isolation, then, this is no different from happening to have a Brass Dragonborn in the former case and happening to have a Red Dragonborn in the latter case.
I think this is a nice way to make a Dragonborn more versatile and tactical. It will make them a bit more powerful, certainly compared to other types of Dragonborn (if, say, you decide only to do this with fire breath, since RAW none of them have cones of lightning, for example, but those fine details are up to you), but I don't think it will unbalance anything hugely.
I'd say let them choose at the time they want to use it, kind of like how Clerics and Paladins can choose which Channel Divinity to use, but can still only do it once per short rest.
Use spells to turn into a dragon
Using true polymorph you can turn yourself into any creature whose CR is equal to or less than yours. Even at level 1 this would enable you to turn into a dragon with a breath weapon.
With true polymorph, the caster concentrating on the spell for an hour would allow you to remain as a dragon until the spell is dispelled.
Of course, this is not ideal for a few reasons. For example, you are now a dragon without access to your class features and true polymorph is a 9th level spell and thus won't be accessible until very high level (if at all).
Shapechange works almost the exact same way except it lets you also keep class features. But it also doesn't become permanent at the end of the duration.
Wish is also an option
As always, wish is an option. You couldn't duplicate the effects of shapechane or true polymorph because they are too high of a level, but you could wish for a breath weapon or wish to be a dragon. Of course, as this is a non-standard use of wish, the 33% of never being able to cast it again apply as well as giving your DM the opportunity to twist the wish however they like. But it is an option.
Best Answer
Currently, not possible
While it is difficult to prove a negative, the Dragonborn's Breath Weapon isn't a magical effect, so does not benefit from any feats that affect magic.
There are feats that affect Dragonborn specifically, but the only one that relates to breath weapons simply allows you to expend your Breath Weapon's use to create a fear effect.