I would try parallelizing the process. Even running two processes on the same machine might help. Just divide the list.dat file into two, and try this:

perl -MLWP::Simple -lne '...' list-part1.dat > results-1.txt &
perl -MLWP::Simple -lne '...' list-part2.dat > results-2.txt &
[download]

I have a feeling a lot of the running time is being spent waiting for the remote sites to accept the connection and respond.