Since I just went through this with some cooks in game, I thought perhaps I should post some clarifications here.

There are only two uses for batagreens -- The Trader in the Forest quest, and cooking Batagreen Sautee. Doing the quest (which is NOT repeatable) gives you the one clump of batagreens you need to finish it, so unless you cook there is no reason to get batagreens any other way.

These are EX, so if you need to obtain them with the character you cook on. You can't sell, trade, bazaar, send or auction them.

Once you get batagreens from Phairet in the quest, you can buy all you want from him...one at a time...for 50 gils (I hate the new term "gils"!) He doesn't have a buy/sell window, just trade him the 50 when he asks. (Trade him more and he'll still just take 50 and give you one batagreen.) They stack so you can buy a lot at once...just one by one. This does NOT change if you complete the quest (you get a Robe in exchange for your greens, which is worth a good deal more) you can still always go back to him for more greens when you need.

At least on Cerberus, Batagreen Sautee (and Bataquiche, which is made from the sautee) just don't sell. Even when they're requested for Guild points they sell poorly.

If you need a level 52 item to skill up cooking, Crayfish Balls are simple and salable.