If a Kobold1 Fighter takes Dragon Disciple and then Breath of the Dragon, its DC falls behind quite quickly2, as you only get trained in arcane spell DCs.
The only way I see for increasing it is also taking Sorcerer Dedication (which you cannot do until you took at least another feat from Dragon Disciple), and then Expert Sorcerer Spellcasting after Basic Sorcerer Spellcasting at level 12.
This is 4 feats for Expert, a hefty price. Are there simpler solutions?
- for access to this Archetype
- assuming it was not abysmal to begin with, Charisma is usually not a focus for Fighters
Best Answer
Not Outside of Archetypes
Your only hope here is through Archetypes, and you've already pointed out the hefty price to get to the Expert DC there. If you're playing in a Free Archetype game, or playing the Strength of Thousands AP (which grants Free Archetype in either Wizard or Druid), your GM may waive the "one archetype at a time" requirement, so you'll be able to get to expert/master DC a little easier. Otherwise, the goal is to make the 4 feat investment worthy in otherways.
Other Archetypes
You mention Sorcerer as a way to do this; any multiclass archetype with expert and master spellcasting benefits for arcane spells can also serve. That's also Summoner for Charisma based casting (note, Summoner archetype... isn't great). If you happen to have 14 Intelligence, you can grab Magus, Wizard, or Witch. Magus might actually have some synergy with Fighter, Witch gets you a familiar, Wizard is probably best for your Strength of Thousands games. But other, non-multiclass archetypes in order of interest:
And, well... that's it. Or is it?
Dragon Disciple May Supposed to be Innate
It seems to me that Dragon Disciple might supposed to be all innate spells. This may seem minor, but it's quite important - if indeed it is innate spells, then any Spell DC we get greater than expert (not just Arcane!) bumps our Spell DC for Dragon Disciple. This is really important because it opens up the archetypes a lot. The evidence? There are three feats that grant you innate arcane spells:
The dedication doesn't actually grant you any spells, and the two other spell-granting feats both grant you focus spells from the Sorcerer bloodline. It seems a little off that some are innate while others aren't, and an argument can be made that the focus spells were also intended to be innate. Obviously, this comes down to GM Fiat, and RAW Dragon Disciple just grants you some innate spells and some non-innate spells. It seems pretty fair to rule the Dragon Disciple spells are innate, though, and given the investment required before that makes a difference, it doesn't seem overpowered by any means (especially with your build).
If your GM rules Innate
Beast Gunner and Sorcerer were really the only ones worth considering before (unless you grabbed 14 Int for other reasons), but this adds many really good options. Since it's literally any archetype that increases any Spell DC to expert, I won't list all the possibilities, but just a few important ones:
TL;DR:
Strongly look at Beast Gunner, and Oracle or Bard if your GM rules that Dragon's Breath counts as an innate spell. These each may be worth the investment, even without Free Archetype.