Hello freelance
I think part of the trouble (which a lot of us suffer from based on posts on here) is imposter syndrome. It sounds like you didn't think you'd get it....and then you did. A bit like how you thought you wouldn't be able to double your fees and then you did!
You are clearly great at doing what you do and also selling yourself. It seem the thing that is out of line is your view of yourself/confidence. Some of this is probably down to the fact you are new to freelancing, it takes time to get used to it and to build up experience of doing things in a freelance way and gaining confidence that it will all be ok. But it could be that you need to adjust your thinking about yourself too - sorry if I am way off there.
Have you read Lean In ? By the female COO of Facebook. It is full of examples of women who are too apologetic, don't push themselves forward etc. I keep thinking "Oh God I do that".
If you don't want to read the book, watch her TED Talk for free!
It does focus on senior employed women to some extent (but we shouldn't forget that we are the CEOs of our businesses, or rather "A freelancer" as I rather self-deprecatingly say which I need to stop!) but the general message that women need to be more confident & fake it until we make it is universal. The book is REALLY challenging my thinking - I think it could be life changing!
I do also think that in freelancing more than employed life there is a feeling of "the time is now" when somebody approaches, so I have lots of empathy with your view there.
I also personally feel that these are my best years (touch wood) and that I need to work now whilst I can etc. I am not aware that of any illnesses but I want to make the best of the best of me etc. That isn't to say that I won't still be working at 70 etc, but I imagine one will need to slow down a little etc.
Only you know how many hours you need to spend with your DC. As Princess says, why not try it? I know you don't want to let the cliet down further down the line, but that is part of the deal of them using a freelancer and a number of things could come up that changes your situation. See if it suits you and the DC and if it doesn't change - that is one of the benefits of our life 
.
I have also found this MN thread about high earning mothers interesting and inspiring.
Mostly they cannot work part-time as their work responsibilities won't allow that. So they have to put their children in childcare full-time and most seem comfortable with that. Not saying that you should feel like that or that your DC should go into childcare more, but I just found it refreshing to see working viewed in such a positive way.
Anyway, I think that is enough from me! I hope you find peace with your decision, enough to try it anyway. And congrats! I do think you have graduated from freelancenewbie - do let me know your name change though, I have enjoyed our chats!