Do you actually want to stay in Miami and that's the reason you are forcing the surface sector MIA-FLL?
You could just as easily do a CPH-EWR-FLL-XXX routing or similar via ORD if you want to avoid the surface sector. This would be no more complex than flying CPH-FRA-MIA/FLL-XXX
Another alternative to avoiding the surface sector is available to the larger destinations in the Carribean such as NAS, you will find direct connections from MCO, CLT even IAH and EWR on certain days, although backtracking might be an issue here, not sure. One could even do CPH-EWR-NAS and be done with 2 sectors.