Using Facebook SDK and retrieving unique application facebook ID's

I am trying to make an app in which a user imports their friends from Facebook into my app. The user of my app would then select 2 of their friends, a match request is sent to the people in facebook, and if they both accept they download my app and a match is made in my app. However, according to my research, this isn't possible because they both would have needed to download the app before a match is made. This is due to restrictions on unique facebook IDs to my app, which every app gets uniquely. Is there any way around this?