As you note, you can have co-ordinated plurals:
- You're [Mary and John]'s child
- You're [the McDonalds'] child
- You're [the couple who robbed them]'s child
All use a phrase and the Saxon genitive. The last one is grammatical but awkward enough to rephrase as
- You're the child of the couple who robbed them
This would suggest that any similar structure is grammatical:
- You're [both of us]'s child
And again, that's awkward enough to rephrase:
- You're the child of both of us.
The noun phrase is not changed whichever genitive form is used:
- You're Mary and John's child :: You're the child of Mary and John
- You're both of us's child :: You're the child of both of us
- *You're both of our child :: *You're the child of both of our
The forms with our are ungrammatical because although we's or us's becomes our, this doesn't happen with the noun phrase both of us: the possessive has to apply to the whole phrase, not simply the last word. In order to apply it to the whole phrase, it's necessary to use the Saxon genitive just as with the couple who robbed them — that doesn't become their.
Using our actually divorces that possessive from both, so that both refers to the following noun (so child is obviously wrong, because both requires a plural).
- *Both of our child
- Both of our children
Because of the ungrammatical nature of the example, it's necessary to change it slightly to produce something which is grammatical, demonstrating that both refers to children:
- You're both of our children, and we never had any more.