Frankly, you should be SRing for Natures 1 at a time as they come in. I'll be honest, I don't know how the pick-up procedure works if she has more than 1 distribution Pokemon, but the point of SRing would be to check 1 at a time then save the one you want. If you collect multiple Pokemon at once, and only 1 has a favorable Nature, you'd lose out on the others, wouldn't you?
Doing multiples at once is at best counter-productive, and at worst a downright waste.