I have had some success with kiss my face products: the olive oil bar soap and the liquid pear soap, even though it has fragrance.
I think the fact that the oil keeps the skin from drying is part of why it works.
Also, this may seem strange, but the brown herbal conditioner from nature's gate is soothing to my condition. I apply and rinse it in the shower sometimes, but have recently tried applying it very lightly and leaving it on after I've washed and dried my crotch with a blow dryer.
Hope this helps some!
Choose a body wash that contains no perfumes, no additives.. try an organic body wash, unscented. Usually even a Dial body wash works best. Or a Provon wash could work as well. Oat and milk help too. Yogurt body washes could be beneficiary in the pH balance of the vaginal skin, preventing yeast infections that can be caused from lichen sclerosis often.