Buon Giorno:
I am not an expert and I am not Italian, but I have a couple of years of Italian study.
Carmine would be pronounced as CAR (like the automobile) ME (like yourself, as the i is pronounced as we say e) NEH (the e is said as and English "a").
So Carmine is Car-me-neh.
Hope that helps and if I am wrong someone please correct me.