So is there no way to compile C++ stuff with Clang MSVC without using Microsoft's proprietary C++ runtime?
Not that I'm aware of, no. Feel free to work on reimplementing what's missing though :-)
Microsoft did offer some parts of what libc++ needs under a suitable license, so that libc++ could reduce its reliance on MS STL (parts about the exception classes, IIRC), but that bit hasn't been integrated yet, and I would think that more parts are required to make it entirely standalone.