I would always allow strength powers to boost breath weapon damage, always. (Though I'd consider allowing a cheaper upgrade that only boosted breath weapon damage if it didn't make sense for the character to also be super-strong. But that wouldn't stack with benefits from super strength.)
I would not allow breath weapon to make a character count as armed for purposes of defense, though gaining that status would be within the scope of a mortal stunt.
I would also, given appropriate aspects, allow a breath weapon power to operate off of fists or guns skill without needing extra refresh to be spent.
Mostly the reasoning is that breath weapon is meant as a flexible power for any sort of innate ranged attack, from actual draconic fire breath, to a master conjurer's ability to call up a handful of ectoplasmic throwing knives whenever she needs some. Strength powers obviously should benefit the latter application. And it's my opinion that a thematic change (conjured knives to blast of flame) shouldn't have the level of game-mechanical impact that losing access to strength boost would represent.
Of course, the other part of my reasoning is simply that breath weapon is a fairly weak power, generally not worth the two refresh. For the same price you could get channeling, which allows fewer attacks but is much stronger & has an actual upgrade path. Or you could get inhuman strength and throw pebbles at people for the same damage output; sure, that won't work if you're in a clean white room with nothing on you, but the breath weapon won't work if you're in a circle, and either way you're probably getting a fate point for your troubles.
Also, Lawgiver: Breath weapon is explicitly not an AoE! It's a single target attack or maneuver. Allowing it to be AoE would be a major upgrade to the power as written.