If there is a papertrail that somehow links to anything American, then at some point a US government agency will be involved. Giving a FF# provides confirmation that you were a flight going to and/or leaving Cuba. The Feds take this very seriously. Even though actual prosecution is rare they can make you sweat while you're in custody and make your future international travel a nightmare.

Its not worth the 5,000 miles.

Off topic, but the same goes for using credit cards or making phone calls while you are in Cuba. Anything that leaves a papertrail with an American company is a potential problem if the wrong person decides to snoop.