I can't believe I missed this (well, okay, the message only showed up after trying to recompile a zillion times and deleting/changing files), but there was a warning about being unable to resolve a reference to System.Web.Extensions.Design -- the project didn't have a reference to this despite having a reference to System.Web.Extensions and being compileable before.
After re-adding the reference, the website compiles. They should really promote that warning to an error!