Is it possible, using official material only (i.e. not homebrew, although if Unearthed Arcana has something, I'd be interested in that, but ideally official material) for one to gain a recharging breath weapon (i.e. not something that's a one-time use like a potion of fire breath) besides the following:

• Being a Dragonborn
  


• Learning the dragon's breath spell from Xanathar's Guide to Everything (not strictly a recharging breath weapon, since it's actually a spell, but I thought I'd include it anyway)

Are there any other ways to gain a breath weapon? I don't mind what kind of damage it does (if there are ways that force you to have a breath weapon of a specific type without letting you choose).

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.

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.

Reflavor damaging spells

From a mechanical point of view, a breath weapon is an effect that deals elemental damage in a cone or line area.

There are a number of spells that do area damage in cones or lines that can easily be reflavored to represent a breath weapon. Some examples include:

• 
  Burning Hands (15 ft cone, fire)


• 
  Aganazzar’s Scorcher (30 ft line, fire)


• 
  Lightning Bolt (100 ft line, lightning)


• 
  Cone of Cold (60 ft cone, cold)

It should be noted that this does have some mechanical differences from a dragonborn/dragon's breath weapon. Namely, being a spell and a magical effect, and thus affected by things such as counterspell or antimagic field which a normal breath weapon would ignore.

Mystic

This is Unearthed Arcana which is not ideal as per the question requirements but the Mystic, as part of the Corrosive Metabolism Discipline (p.12) has access to two breath options:

Breath of the Black Dragon (5 psi). You exhale a wave of acid in a 60-foot line that is 5 feet wide. Each creature in the line must make a Constitution saving throw, taking 6d6 acid damage on a failed save, or half as much on a successful one. You can increase the damage by 1d6 per additional psi point spent on it.

Breath of the Green Dragon (7 psi). You exhale a cloud of poison in a 90-foot cone. Each creature in the line must make a Constitution saving throw, taking 10d6 poison damage on a failed save, or half as much damage on a successful one.

Given the psi limits this requires a Mystic level of 5 for the Black version and 9 for the Green one which is admittedly a pretty heavy cost if all you want is access to the breath.

While that doesn't exactly fill the no homebrew requirement, if the game is not official (i.e. Adventurer's League) anything you can think of and DM allows is on the table. Your imagination (and DM willingness to go along) is your limit.

You can ask your DM to let you just do it for a backstory of being some kind of scientific experiment. Honestly, it's not that big a deal to be able to breathe fire like a dragonborn does, I as a DM wouldn't want a big payment for that - I'd probably just let it be a custom background feature like standard backgrounds have.

If you want to do some more work you can have something happen, through science or magic, during game. Again, the details are endless, but I already mentioned experiments. I can also think of some nice scenes involving a fight with a villain, where the Dragon Mask you mentioned breaks and through some twist of fate the character gets it's powers embedded in him.

While those aren't really "In-rules" ways, I don't really consider them homebrew, because those would be a one-time thing, and DnD is a lot about giving players what they want - twists of fate happen a lot in adventures, and there are some unique things that can happen as part of the game even in official adventures - Wand of Orcus immediately attuning itself in Out Of the Abyss comes to mind, but there probably are other examples.