Is the phrase "for one of both of us" grammatically correct?
Your reasoning (You cannot treat "both of us" as a list of two alternatives because 'both of' explicitly means they are not alternatives) is correct.
There is no syntactical explanation. "One of (the) both of us" is as grammatically sound as "Colourless green ideas sleep furiously". It is the semantic content of the words that makes both them unacceptable.