stayorgooo you don't need permission not to go, you want to go, you are sorry not to but you cannot manage alone and dh will be with your sick child. I think you really need to just tell her once you decide.
Chicken pox can be quite serious, I am sure it will not for your dd, but it can be so should be taken seriously. And you obviously have mobility issues which cannot be changed. Don't feel bad. If you are worried about how to tell I would email with all the details and then say I will call tonight as I really want to chat to you and I am just so sorry to miss your special day.
Hopefully your friend will totally understand but if she does not, really, you must do what you feel able to do. You cannot do what you are not able to.
At least by telling her now she may choose to invite someone else (don't suggest it, but she might!)
Try not to feel bad, it is not about how much you like or respect your friend it is practical and no one else knows how ill your dd feels or how difficult it is for you to manage public transport.
Personally I would also be ready just in case she tells you she has a friend driving all the way there who can give you a lift, if she does, what will you say, be ready in case! Unlikely but possible.
If that were to happen would you take up the offer? If not, I would add into your not being able to travel by public transport that you also do not want to leave dd as she needs you etc. This may be totally true but if you say it afterwards it may sound like an excuse!
Goodness, I sound devious, I guess I just over think things. And I was the kind of bride who would have ferreted out a lift for a friend! I actually invited a virtual stranger to my party who I met a few weeks before (lonely foreign student type) and she ended up being a pain, needing a lift home and appearing in loads of photos! I am not even sure I remember her name now!