This smells like it may be a routing persistance issue rather than an error in your code.   How do LWP::Simple or core Perl know internally that the active network interface and default gateway have changed mid-stream?   Not that they couldn't necessarily be made to refresh - I'm just asking, eh.

One possible (potentially hack-ish) workaround would be to automagically kill and re-launch your script using your Unix distribution's standard mechanisms.   With Debian Linux that would be scriptlets added to /etc/network/if-up.d/ and if-down.d/ directories.