I have been making cold-processed soaps for about a year now. I have made 60+ batches now. Be forewarned it is addicting, and can be a VERY expensive hobby.
Soon after I started soaping I was begged into making candles, then lip balms...and so on. I now make dozens of toilety items. It is addicting, and FUN!
For the Mama with allergies in the family...aloe vera can also be an allergen, as well as the peanut and lanolin oils. I would start off using 2 base oils at a time and NO additiives or fragrances. Maybe try adding one at a time.
I have made soap with ONLY 1 oil and lye as well. Actually several one oil batches. I tend to like the ones with more oils though. Adding beeswax to an all olice oil soap will help it trace and harden more as well. I have a beekeeper in my town, so, mine is all local, local honey can actually help with allergies...check into it.
Also, some soapers will not use beeswax as it can clog pores.
I will have to look up amounts for you (sleeping babe on lap), but, I have a great recipe with just vegetable shortening, olive oil and beeswax.