I remember this question popping up a lot on the forums (And eventually getting an official answer) when I was still playing D20Modern.
The answer is that they do not stack - in any fashion. Not even to the degree DuckTapeal says in his post. Every time you roll to attack somebody, you must decide to either Brawl or use Combat Martial Arts.
The word from the developers on why two fighters couldn't win against each other when only Brawling (For combats where characters have sufficiently high HP and CON values), the answer was that in most brawls, you'd expect a combatant to eventually pick up a bottle or other means of dealing real damage.
SOURCE that brawl and CMA don't stack: https://www.wizards.com/default.asp?x=d20modern/bp/20030320a
SOURCE that Advanced CMA and Knockout Punch don't stack: https://www.wizards.com/default.asp?x=d20modern/bp/20051122a
SOURCE that Street Fighting does stack with CMA: https://www.wizards.com/default.asp?x=d20modern/bp/20060411a
The short and not very helpful answer is that both of these builds have far too little damage for 10th level play if your goal is to deal damage. I would not consider either of them "respectable" contributors in the damage department, though the starknife build clearly edges the shortbow build out in calculations.
In both my example calculations below, I assume a +2 enhancement bonus weapon and the bard casting Heroism on himself (a long duration buff that I feel is reasonable to assume. Other buffs may apply as well).
Starknife variant
Your attack roll is: +16/+11 (+7 BAB, + 5 CHA, +2 ENHANCE, + 2 HEROISM)
Your damage roll is: 1d4+7+3d6 (+5 CHA, +2 ENHANCE, 3d6 sneak when applicable)
Against an average CR10 foe (AC 24), this equals a full attack DPR (damage per round) of 10.97 when not sneak attacking and a DPR of 22.00 when always sneak attacking.
Shortbow variant
Your attack roll is: +14/+9 (+7 BAB, + 2 DEX, +1 POINTBLANK, +2 ENHANCE, + 2 HEROISM)
Your damage roll is: 1d6+6+3d6 (+1 POINTBLANK, +2 ENHANCE, +3 ARCANESTRIKE, 3d6 sneak when applicable)
Against an average CR10 foe (AC 24), this equals a full attack DPR (damage per round) of 8.88 when not sneak attacking and a DPR of 17.81 when always sneak attacking.
Since sneak attacks are easier to pull off in melee than ranged, the Starknife's flexibility will pull it further ahead too.
Note that a CR10 enemy has about 130 HP, so neither of these two builds is taking less than 6 full rounds under perfectly optimal conditions (always full attacking, always sneak attacking) to down what should be an equally matched enemy.
CR10 enemy stats here are taken from the Monster Statistics by CR table provided by Paizo.
Best Answer
Normally, when you multiply damage, extra bonus damage dice, like sneak attack or flaming, apply only once.
But Desna’s divine fighting technique doesn’t multiply damage. It says you score multiple hits, which is not the same thing even if it is off of one attack roll. It then mimics damage multiplication for the purposes of precision damage, but neglects to mention any other source of bonus damage dice—like flaming. And then its says “other damage bonuses” all apply to each strike, i.e. are “multiplied” (but technically aren’t).
So the written rules say yes, you get bonuses like flaming repeatedly with Desna’s advanced technique. Do I suspect that whoever wrote this misremembered the actual damage multiplication rule, and meant to say that all bonus damage dice don’t get multiplied? Yes I do—but I can’t back that up.
Anyway, it doesn’t really matter because you shouldn’t use Desna’s advanced technique anyway—even if you can apply flaming repeatedly, even if you could apply sneak attack repeatedly, it’s completely without purpose. Any character that qualifies for it and has any business making weapon attacks1 has far higher expected damage with a full attack than with this technique, so don’t bother with it. Paizo shouldn’t have bothered to print it.