I would take a break from the elderberry syrup to make sure that this is the cause. If it truly is the elderberry syrup, try decreasing the dose until no diarrhea is occurring, if this is too small to be therapeutic then it may be that the person experiencing this has a sensitivity to elderberry and have to avoid it.
You could try to reduce the irritation/laxative effect with astringents or digestive supporting herbs like meadowsweet, raspberry leaf, chamomile, calendula, plantain or ginger, however that is a pretty low dose for that outcome.
I think it may just be a matter or testing out a few dose reductions and other herbs with it to see if this helps. If not, it may just need to be avoided and other antiviral/immune supportive herbs or mushrooms used instead.