Hacohen because, as old man said – it’s unnecessary and irrelevant, Harav – because he’s in the need of rachmei Shamayim – not the time to show off “airs”.
I believe Sefer Chassidim says that when one is davening for his father, he should not say “Please heal my father ploni ben plonis”, he should just use the name. Perhaps the reasoning behind it is the same for a kohen?