I could see a GM allowing Fickle Attack to cancel a Dragon's Breath misfire... or applying both. That is, you treat the 1s as max for damage done, but then still suffer the effects of a misfire.
Fickle Attack (Ex) (Mythic Adventures pg. 48): Whenever you roll damage for a melee or ranged attack with a weapon or alchemical item, you can treat any natural 1s on the damage dice as if they were the highest possible number on those dice. You can select this ability up to three times. The second time you select it, treat 1s and 2s as the highest value. The third time, treat 1s, 2s, and 3s as the highest value.
Dragon’s Breath Cartridge: (Source Ultimate Equipment pg. 41) This cartridge contains alchemical compounds that, when fired, produce a cone of fire instead of the normal attack of a one-handed or two-handed firearm with the scatter weapon quality. The nonmagical flame deals 2d6 points of fire damage to all targets within the cone of the scatter firearm (DC 15 Reflex save for half). These cartridges cannot be used in firearms that don’t have the scatter weapon quality. Because this ammunition forces a saving throw instead of making an attack roll, the misfire rules are slightly different. If you roll a 1 with either of the damage dice, the firearm misfires.
It's not an attack, so I guess it hinges on if it's an "alchemical item". It's "alchemical", but not made with craft(alchemy), so it's questionable. I could see a GM going either way on this, but it's MYTHIC, so I'd probably lean towards allowing it, since it's not that OP compared to a lot of mythic stuff.