Based on parses from the Moogle fight.
A comparison of Whiskerwall, Pukna, Pukni, and Woolywart. It appears based on our initial limited testing of 3 of our linkshell's archers that Attack Power has a significantly higher impact on damage when the mob in question itself has a high defense.
Archer 1:
Dexterity 287
Piety 233
Strength 215
Attack Power 531
Crit Attack Power: 34
Archer 2:
Dexterity: 315
Piety: 267
Strength 220
Attack Power: 502
Crit Attack Power: 0
Archer 3:
Dexterity: 316
Piety: 242
Attack Power: 502
Crit attack Power 36
Average of 3 parses (may do more testing now that we have the fight on farm status, however we are not guinea pigs our gear will probably have changed by the time I test again):
Now based on this it appears that Archer 2 comes out ahead. Supporting the hypothesis that DEX/PIE > ATK for endgame archers. Note the inconsistencies however, Archer 1 retains comparable damage and has top damage in certain areas. What's more interesting is that Archer 3 also shows very good damage in many areas, despite having a lower overall damage (and lower stats for the most part, other than the crit attack which could explain it).
Still overall looking at this, and as I'm sure many of you have done looking at full encounter parses (not filtered by individual moogle) I'd guess that an archer's priorities should be DEX>PIE>ATK in that order. Though given the wide damage ranges, it is hard to say for certain.
Now let's look at Whiskerwall on the same 3 fights:
The results here are far more clear, with Archer 1 seemingly having a significant advantage in many areas due to his higher attack value. Again the damage range plays havoc on consistency.
Now as I said these tests are extremely limited, and while I'm thinking right now that endgame Archers should focus on ATK>DEX>PIE (which really boils down to choosing Heaven's Fist on your gloves) I wouldn't recommend you go out and spend millions you can't afford just yet.
What I will say is this though, testing on normal mobs has extremely limited usefulness if it is true that Mob Defense and Player Attack are checked against each other, We have also tested and there does appear to be a damage cap in place (using lower level arrows and increasing stats).