Keep in mind that Chicago is now a FAFSA only school and Georgetown still uses the profile. So, off the bat they gave different methodologies of calculating your need. Did you read Georgetown policy in appeals or financial review
Georgetown only offers need based financial aid (they call their financial aid scholarships). So they will not care what you got in merit from USC, they are not going to “match it”. But you could have found this out from simply reading their financial aid page;
requesting additional aid
Most schools are upfront with the fact that they will consider an award letter from a peer school.