My story is very similar to yours. So, I wish you can get a good result on your application!
GMAT should not be a big problem as long as you can get good connections with faculties in that school, either by yourself or through your reference letters.
My 2 cents worth - while a high GMAT will not necessarily get you in, a low GMAT is likely to get you out. With a 680, your profile is unlikely to be competitive for a top-20 finance programs (unless you have very strong and credible LOR, published papers in top journals, etc). Where you graduate from and with what honours don't really matter much, because the next applicant is likely to be from a even better school with higher honors. 一山还有一山高.
If you still want to stick to your current score, I suggest you widen your scope to lower-ranked schools.
I know someone with GMAT 770, GPA 3.9 - and he still encountered challenging times in getting into a top 20 programs. I'm not trying to discourage you, but that's the reality.