I replaced body soap/deodorant/lotion with coconut oil about 6 months ago, and find it very effective for staying clean and odor-free - even during hot San Antonio summers. It's supposed to have great anti-fungal/anti-bacterial/anti-viral properties (which I suppose is why it works so well as deodorant), so I would definitely give it a try on your rash.
I'm new to going poo-free, though. I'm on day seven without shampoo, and up until yesterday, things had been tolerable. A bit greasy with daily baking soda/vinegar "washes," but tolerable. Last night though, my hair felt like I had poured candle wax in it - I literally could not drag my fingers through it. :p Every time a gust of wind blew it, it stuck in that position. Ew!! I really didn't want to throw in the towel after I've come this far, but there was NO WAY I was going to work looking like that. :(
I did more research about what others - women, specifically - are doing, since I doubt this icky, waxy stage would be much of a problem if my hair was less than an inch long like my husband's (who successfully went poo-free over three months ago). Definitely a mixed bag of experiences, and it seems like it may have something to do with how often you used to wash. In the last few years, I'd cut back from washing every single day to every other day; but by the morning of day three, I'd have to wash, thanks to excessive oil. So I guess my scalp's oil glands were used to being stripped fairly often.
Quite a few people suggested doing the "conditioner only" (CO) method while you're adjusting. Apparently, they contain very small amounts of surfactants, so they will get rid of some of the oil without stripping your hair and scalp. I've heard of girls with super coarse, curly hair doing this, but it never occurred to me that it could work for fine, straight hair like mine. It seemed to me that conditioner would be counter-productive, since it's just more oil... but I was kind of desperate and I tried it this morning. SO much better!! The trick is to wet your hair and distribute the conditioner as soon as you get in the shower. Massage it into the scalp really, really well, using your fingertips. Go about your other cleaning rituals, and don't wash off the conditioner until you're ready to get out, so it has a bit of time to sit on your scalp. Rinse really, really well, with more scalp massaging. You also have to make sure the conditioner is silicone-free, so there's no possibility of build-up.
So today, my hair is by no means free of oil - it's still heavier than I'm used to dealing with, but I don't think anyone but myself could tell that I haven't shampooed since last Tuesday. I'm starting to think that the squeaky clean, light-as-air hair I'm used to is not natural at all, since that implies that it's also completely devoid of all natural oils. I hope I can eventually get off commercial conditioner too, but for now, this should make the transition bearable. I'm going to give the egg "wash" you mentioned a try sometime soon - that would definitely be preferable to commercial conditioner, if it works for me.
This experiment has also inspired me to weed through my facial cleansers - turns out there were a lot of sodium sulfate ingredients in there - a harsh detergent also used in dish soap. :( I tend to wash my face 2-3 times a day because (wait for it...) I get too oily! Hmmmm... I'm tempted to try washing with an oil/grain combo (like coconut oil and corn meal or something) and see what happens.
This stuff really opens your mind to a whole new way of thinking and questioning modern "innovations." I'm starting to think that word is just defined as "ways to make money."