Technically, Epic Boons are completely official (though optional), and can do all sorts of things.
This would be for particularly high level play only, and entirely at the discretion of the DM, but Epic Boons are basically "DM hands you some appropriate power that fits your character and develops your story." There is no Epic Boon specifically about dragon breath, but they're specifically called out as the sort of thing the DM can/should customize.
As such, granting a dragonlike breath weapon to someone who had as part of their story seeking to be more dragonlike would be entirely in keeping with both the formal rules and the intent of the feature.
Epic Boons are in the DMG, pg. 231-232
Potion miscibility is a funny thing.
There are (optional but official) rules for mixing potions, which include what happens when you drink one potion while under the effects of another. There are all sorts of possible effects (including "explosion" and "deadly poison", both of which are kind of unfortunate if they happen in your gut) but one of the (rarer) possibilities is that the effect of one of the two potions is rendered permanent. What exactly that means for any given potion might require a bit of adjudication, but it seems reasonable that rendering a potion of dragon breath permanent might give you a rechargeable breath weapon (talk with your DM). Of course, there's a complication there, because that means that you're constantly under a potion effect from then on, which means that you roll that funny, funny table every time you drink any potion from then on. If you really crave that breath weapon, though, and you have friends who can bring you back from the dead as many times as it takes, and you can buy a whole mess of dragon breath potions to run the experiment on....
The rules on mixing potions are also in the DMG, in a sidebar on page 140.