Actually just went out to get a reference and it sounds like you can complete an exotic sword and then if you start the exotic sword quest with a different sword on the same char it will start at the resources:
Just like the title says. I finished the exotic sword quest on my hunter then started it with my other two swords on the same character, and it started me at the point where you get ability kills and elemental materials!
Clarification: Build your sword on each character then transfer it to the same character before infusing it to 280, which activates the quest for the exotic sword.
Update: This did work without any hitch for all three sword questlines. I have all 3 of my exotic swords now! (Save yourself some pesky time in the crucible!)