I don't see a problem. First, you're dealing with Stephen Foster, and "Doo day day" is absolutely in place, and doesn't call for any other reference.

I've never heard "dah" as meaning "trinkets".

Yes, if one is looking for other meanings, "Doo" could be taken by itself as referring to a hair styling, or a "do rag". But in context that would not suggest itself, I think. And if it did, it's just meaningless.

I can imagine someone saying "doing the do" for sex, though I've never heard it, myself.

And as to being characterized as babyish and uncool, I think not, especially in the context of Foster.

Dave Oesterreich