I don't know if you are allowed or not, Expatkat. They do want to have evidence that you (the parents as well as the child) are legally resident in the US - green card, US passport, but they don't go as far as asking for a utility bill or anything like that. I'd say that if you are on holiday in the US, then you shouldn't be getting a passport from the Embassy (and it's horribly expensive).
I'm not totally aware of your situation, but from what I can see, it appears that you are soon going to the US for a long holiday and will be returning to the UK afterwards. You are all going to the US on US passports and you are worried about your child being let back into the UK as a resident, given that his British passport will be out of date at the time.
Do check with the IND, but I would be surprised if he were denied entry into the UK - your PR stamp (or British passport, if you have one) and his expired British passport should be proof enough that he is resident here (it's also worth carrying the documents that entitles him to a British passport - your passports, marriage certificate and his birth certificate). You'd get a small slap on the wrist, though. Saying that, I cancelled plans for a day trip to Calais last Easter holidays when I discovered at the last minute that my daughter's UK passport had just expired.
BTW, it takes less than two weeks, by normal processing, to get a British passport. Fill out the form on the Passport Agencies page, and you'll have a printed version the next day. Beyond that, you just collect what you need (photos, old passport) and line up a counter-signatory and you can put it straight back in the post. Because you entered the data yourself, they do minimal processing to actual produce the passport.