Eep- that's about my worst nightmare. Sorry you're going through this! I have to say, I think the home office is right- you need to get this sorted ASAP, and I wouldn't personally chance leaving the UK without a new visa. The trouble is that even if you did want to pay the £500 for the TOC stamp in person, you may not get an appointment by mid-January- I made an appointment for my IGS visa a couple of weeks ago, and the Liverpool office was booked up 4 weeks in advance.
I'm not sure if there's a way to sort this out in the UK without canceling your trip...you'd probably want any new visa to be in a regular, non-emergency passport anyway. If I was in your situation I'd cancel/reschedule the trip if at all possible, get the passport replaced and then post off the NTLTOC form (which is only £160 by post, much kinder) and hopefully have the new visa within the next month or two. Straightforward, done and dusted. But if you're really looking to go on your mid-Jan trip as scheduled, I think it'll become a lot more complicated. 4 days isn't really enough time to get a new visa in the US, unfortunately...