Gathering is definitely key for BTN/MIN/FSH retainers. I've got a couple of retainers with perception gear also equipped and a couple without, there doesn't seem to be a noticeable difference how many HQ items I get when I'm doing the BTN/MIN/FSH ventures.

I wouldn't bother going for HQ unless the item has gathering. And it's helpful to meld cheap Guerdon materia where you can.