Visual Studio 2012 can't build on Windows 8 due to conflicting rc.exe with SDK 7 & VS2010

Question

I have a Windows 8 machine running Visual Studio 2010 and 2012. The Windows SDK 7.0A was installed (presumably during the Visual Studio 2010 install) and I've recently installed Windows SDK 8.

I need to have VS2010 Pro and VS2012 Ultimate on my machine because I'm working on team projects that tie me to both.

I can successfully build my solution in VS2010 and in VS2012 if I leave the Platform Toolset set to v100. But if I change the Platform Toolset to v110 in VS2012 I get a

LINK : fatal error LNK1158: cannot run 'C:\Program Files (x86)\Microsoft Visual Studio 11.0\VC\bin\x86_amd64\rc.exe'

As suggested in answers to similar VS2012 RC.EXE questions I have copied RC.EXE and RCDLL.DLL from the Windows SDK 8 folders to the MSVC11.0\VC\bin folders putting the 32 bit versions in the bin folder and the 64 bit in the bin\x86_amd64 and bin\amd64 folders. None of these folders is on the PATH.

I can cut and paste the error message rc.exe reference and run it from a command line so I know that it's there and it runs. It's Version 6.2.9200.16384.