I managed to get a room through the Travelclick site. I've been getting "Access Denied" notifications on both websites for close to an hour now (and still am, just checking to be sure), but I still had an active session open on the Travelclick one from earlier, and after I changed some dates (checkout date changed to 20th instead of 21st) it suddenly worked. Got a confirmation email and everything.
So yeah, that took almost 6 hours, and I get to stay a day less than I was planning but, honestly, I absolutely dare not complain right now with the mess everyone seems to be in...