Something similar I often do in CGI programs and other utilities that are not attached to the terminal:

BEGIN {
  open STDERR, ">", "/tmp/$0-$$.log";
}
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and then all die and warn messages go to the log file in /tmp.

To get this to work with Sys::Syslog you would need a little more machinery:

BEGIN {
  sub SLH::PRINT { syslog("err", $_[1]) }
  sub SLH::TIEHANDLE { return bless ["dummy object"] => "SLH" }
  tie *STDERR => "SLH";
}
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And then everything written to STDERR gets passed along to syslog.

Disclaimer 1: I didn't actually try this.

Disclaimer 2: I never, ever use software written by Eric Allman.