Welcome to the site!! I don't have any children, but hopefully I can help. As far as condish with protein, some curlies experience frizz and dryness with proteins, and try to avoid them when doing conditioner washing or general conditioning. I don't have that problem, but you're daughter may experience frizz or over conditioning.
Products with petro based ingredients I generally use at the bare minimum or avoid all together, unless I wear my hair straight. They can weigh the hair down if used too much, but some people can use it without any problems. Everyone's hair is different, so it's a trial and error process.
With honey, you can either add it to the condish(in small concentrations) or you can mix it in with Aloe vera gel.
Since your daughter is two, I wouldn't worry about using too many products or making the routine to difficult, or worry about curl definition(styling). Her hair most likely will change a bit with age, so the simpler the process, the better. It's really about keeping her hair moisturized and healthy.
With my hair, my mother washed maybe 2 times a month, moreso if I was extremely active, and used a good, but cheap condish. I recommend first some of the Suave conditioners(milk and honey, etc) or some of the VO5 moisture milk conditioners(strawberries and cream, etc). These are cheap, and offer good conditioning. If they don't work, you won't be at a complete loss(only 99 cents) My mother used these products with great success. You may also get away with just using Creme of Nature shampoo(red label). It worked well for my hair as a child(and moisturized enough without having to use a separate condish and detangled well). This can be found in the ethnic section of most stores and contains no petro based ingredients. This is still one of my favorite products.
Also, I very rarely wore my hair down. When wore down, it tangled very often, and made it very difficult to comb, and made me one sour little girl. Her hair needs to be in protective styling......in braids. This protects her hair from the elements as well as prevents the hair from tangling and her ends getting frayed. It's nice to have her hair out once and a while, but I think it should be kept up. For a hair oil/dressing, may I suggest maybe using coconut oil(with maybe some shea butter mixed in). The oil isn't too heavy, and does a good job of providing moisture to the hair. If this doesn't work, you can use both as body butter.
You said that you normally can't leave her braids in for more than a week. That's actually pretty good for an active child. At the most, I could go 2 weeks, but that's only because my mom tied my hair down.
Maybe you can buy a satin pillow case. This helps me when I wear my hair in braids, and is also more gentler on the hair than cotton.
Don't know if this has helped, but I tried.