I'd always found the Bridesmaid's Sandals (and the male equivalent) sold in Ul Dah for 3k or 6k (I forget which atm) to be a relatively good source of FC3's. On average, think I was getting a FC3 about 15, 20 percent of the time.

Now, if that person was skilling up, i'd figure he/she was doing the i180 gear that used to be Red Token but is now Blue Token gear, from Idyllshire. I cannot think of anything else that would be spammable for several desynth classes that could yield Fieldcraft demimateria rather than battlecraft demimateria.