I love, love, love Briar Adeline! However, I do love Adelaide, and don’t see any reason why it wouldn’t go well with the surname. I agree with Marissa that having one name a direct connection would be even better. But either way, ignore your families. It is a beautiful name and such a lovely tribute to great-Grandma.
I don’t mind Briar for a girl or that you changed the spelling but I wish you’d keep Adelaide to be a more direct connection to your grandma. The d/p sounds running together aren’t a big deal in my opinion from middle to last. Either way, best of luck!
Why do people share their name choice before the grandparents are holding their chubby wubby brand new offspring?! Use it. They will adjust, and if they don't then tough. They can call her adeline. As for "don't worry we'll name our future son after your side", that is a mistake in my opinion. What if you only have girls? What if you don't have any more? What if the name becomes the next Isis? Don't make promises you maybe can't keep. Otherwise, throw an extra middle name into this one's name to honor their side.