I'm curious, if you so closely identify with your Jewish heritage, why don't you practice the religion? Judaism has an incredibly righ cultural, religious, (and yes) ethnic heritage, but I would assume embracing the latter a least facilitates if not necessitates acceptance of the two former. I am a quarter Jewish, but have never considered myself anything other than Caucasian because I (like yourself) was never taught to embody all that it means.
It certainly wouldn't have occurred to me in a million years to indicate it on an application 1) because it's not who I am and 2) at the end of the day it's not helpful.