Conflicting run-time libraries can cause serious problems if you are sharing C run time objects (items such as FILE*) or sharing memory allocations (allocating memory in one part and deallocating it in another). I would avoid "forcing" them together to avoid weird problems and crashes.
Instead, if you cannot change the build of the lib or of your dll, I would wrap the lib inside a dll with a C only interface that doesn't leak any C runtime object or require you to free memory outside of the dll that was created inside.