Is his name pronounced Bastache like mustache or like headache or like bast-ak?

Its probs bas-tak

I personally in my head pronounced it as Baz - tash - chay . but the e doesnt have an accent so its from French "Basque" and is pronounced Basta [bas.taˈʁɑʃ] I believe.

i pronounce it as baz-tash. the ch is like sh and the e is silent.