I found the solution — here: https://stackoverflow.com/a/1073772/674976. The distributable DLL can be statically linked to my DLL, thus removing the dependence. This is done in VS 2013 in Project Properties (Alt + F7) > Configuration Properties > C/C++ > Code Generation > Runtime Library, which should be set to Multi-threaded (/MT) from the default Multi-threaded DLL (/MD).
However, I wonder if by doing so I am actually still using redistributable DLLs of two versions and thus risk mixing the heaps and potentially causing memory allocation problems.
If someone suggests a better solution or a clear explanation why I should simply use Visual Studio 2010, I'll be happy to accept it is an answer.