my daughter daisy(2 1/2) also has alopecia she is even starting to lose her eyelashes ( alopecia universalis). she started losing all her hair in january this year. i know how hard it can be ... i went through all the crying and feeling hopelessness thing already (i think i must have laid in bed a few months being depressed. couldn't do a thing!!) and now just recently i've come to terms with it and fully accept it.. and i'm soooo grateful that my daughter feels happy, healthy and can run around and do what other kids do.
i understand how you can feel anxious in public with everyone staring, but my advice to you is just don't let it bother you. people are just going to give you looks of sympathy (well meaning) because they are going to think that your daughter is going through chemo or something and is dying or they are going to keep their children away because they think it might be something contagious. and it's all because they just don't know and are just not educated about it. alopecia awareness often gets pushed to the side because it is not a life threatening or real health issue. so getting back to the point.. if people stare, just stare back at them and make them feel uncomfortable. but also most importantly talk to them so that they understand and are aware that this is just something that you are born with and can't help.
you are going to have to be an extra strong mother... if you accept the baldness, then your daughter will too. and ultimately she will accept herself. at this age they just absorb everything like a sponge and most of what they do or feel is from you... you are their #1 role model!
i agree with the other responses with the big floppy hats as it is a health issue. however if your daughter doesn't want to do hats either, my daughter does lots of hoodies. they make lots of short sleeve kinds with hearts/skulls/bows patterns at target ( everyone just thinks shes so adorable wearing a little hoodie around!)
as for the wig thing i'll let daisy decide if she wants to wear one or not. i know that they make really comfortable ones that don't come off when you hang upside down or even swim! they are called freedom wigs and start around $2800 though. many alopecians wear them without their friends or family even knowing for many years!
i would suggest www.alopeciaworld.com for support so that you don't feel so isolated...you are definitely not alone. there is even a famous runway model anna fitzpatrick who has modeled bald bold and all. she even has a boyfriend who sees her baldness as not that big of a deal. your daughter's life is not limited , if anything it will make her stronger like no one else. embrace it!