Which one of these is more French?

Hello, um so my moms last name (and everyone else on her side of the family) there last name is Grolet, but you see there is two different ways to spell Grolet, there is that way and Gurul� (btw its the same name so dont tell me it isnt, just different ways, TRUST ME I KNOW) but my mom claims that Grolet is more french than Gurul� she says thats the American way of doing it, but i have no idea, i mean my mom does a lot of reasearch so i dont know, i havent looked that far into it, so i figured i would just ask here, so does anyone know which way is more french? (please dont say Gurul� is because of the accent over the "e" because Grolet has an "et" at the end which is very french) btw, its pronouced Grwu-lay thats how it is pronounced, and sorry for all of these misspellings, its 7 in the morning and i havent slept yet...im tired. please i would like somone expirenced in french or who speaks fluent french as a native language to answer this (btw my whole family is french, i just dont want to ask em, they might laugh at me if i get it wrong..)