I know pretty much nothing about windows stuff (internals of a DLL, etc), but when I type file winmm.dll against the one from the midi_rename project and the one coming from Wine itself, I get a difference that may be of relevance:
winmm.dll: PE32+ executable (DLL) (GUI) x86-64, for MS Windows, 6 sections
winmm.dll: PE32+ executable (DLL) (console) x86-64, for MS Windows, 20 sections
Just a guess, but it could be the winmm.dll from the midi rename project is only overriding a few instructions from the whole instruction set of the default one (6 vs 20 sections). And maybe replacing the system32 one with a limited winmm.dll is not going to work as it is missing important sections. I could be completely off the mark though.