Replying to myself one more time, I found, after digging through source code, that the problem here is that you need to add the HttpRequestHandlerAdapter in your WebMVC configuration as well. Once I did that, the JsfResourceRequestHandler worked properly.
Of course, now I've got other problems, because we were using OGNL as a replacement for the EL engine, and the resource[...] notation that the ECSS files use does not seem to work with OGNL. I'm going to try to switch over to SpringEL, and see if that solves the problem, but I think that's starting to get off-topic for this thread, so I won't follow up further here.