Yes, it does. It's very idiomatic. Child me = me as a child / the person I was when I was a child. Adult me = me now as an adult / the person I am now as an adult. So the whole thing means that if the person you were as a child could see you now, they would be very proud.

I've never heard anyone say: "Child me" In the USA, but we do say "Kid me". One other correction: "...would be so proud OF how many..." In English, we are "proud of" things, not AT things. I hope that answers your question. :)

"All I'm saying is that my child would be so proud knowing how many bags of groceries I can carry in one trip."