I take it to be a grammatical utterance, and if it is, then there's no reason why it can't be written down. On the other hand, although this particular construction isn't considered in the Cambridge Grammar of the English Language, other statements there would suggest it's not grammatical.
In English, the 's marks a whole noun phrase (NP) as genitive, not just to the noun, and "the apostrophe occurs as a case marker on the last word of genitive NPs" (CGEL, p. 1763). A very clear example of this is when an adjective functioning as a modifier follows the noun, as in poet laureate. The speech given by such a person is the poet laureate's speech, not *the poet's laureate speech.
Let's take, for the moment, the example the kid who goes to my school. This is like the poet laureate in that it's an NP with the modifier following the head noun. The only difference is that this time the modifier is an integrated relative clause instead of an adjective. Still, the 's attaches at the end of the whole NP.
In your example, though, the relative clause is supplementary rather than integrated. According to CGEL, NPs like Billy constitute the NP by themselves, "but the supplementary relatives do not combine with them to form larger NPs. We suggest in Ch. 15, §5.1 that the antecedent + relative clause here is a special case of a supplementation construction, which is distinct from a head + dependent construction. The supplement is in construction with an anchor (in this case the antecedent), but does not combine with it to form a syntactic constituent" (CGEL, p. 1058).
If the supplementary relative is not part of the NP, then placing the apostrophe after it would contradict the claim on p. 1763.
In this case, though, I think CGEL has it wrong.
[Update: After consulting a number of people, including the authors of CGEL, I haven't been able to find much support for the idea that this is grammatical. It seems like it's something somebody might say to get out of a syntactic bind, but the people that I've asked about it all judge it to be ungrammatical.]
It does, however, raise the question of comma placement. It seems very odd to write, school,'s but I don't see a better solution. The question, though, is just 'can you write it,' not 'how would you write it,' so I'll leave it there.