Ideally no shampoo at all it's not necessary and is drying, with the technique done right conditioner-only washing will remove even a heavy coconut oiling or cleanse greasy teenage hair. The nice thing about it for kids is with the right product every stage is working towards the goal of detangling because conditioner can be much slippier than most shampoos. So for some parents there is a lot less drama at hairwash time. It all sounds a bit strange but it's not any different to removing your make up with a cleansing lotion or someone washing their body in a cream product because they have eczema.
If you really can't get on with conditioner washing after a few weeks you could switch to a super gentle sulphate free shampoo, BUT at that point I would find a way to cover your daughter's eyes. Skin and hair like pH 4.5 to 5.5 (acidic), eyes tend to prefer 7+ (neutral to alkaline) so some products that are actually gentle can sting IYSWIM. There are face shields or you can try the lying back counter wash as per the video linked to in my last post, or of course the curly kids ranges which will include some tear-free formulas.
Frizz and dryness are intimately linked, curlies often need more conditioning than straight hairs so many of us use leave in conditioners. Pure argan oil is a rich source of the fatty acid oleic acid which will penetrate more porous hair, but if her hair is virgin it may not be that porous anyway and if it's fine you might find straight oils weigh her hair down or looks greasy.
It tends to be easier to distribute natural plant oils if they are as part of a conditioner formula or styling product. If you want to purchase argan oil be sure to get pure oil, many brands are actually wall to wall silicones (artificial drying oils that can build up on the hair, need shampoo to remove). Other natural oils that can penetrate hair include coconut oil and avocado oil.