|Keep It Simple, Stupid|
What you've described is a system issue... and you want Perl to point out what's wrong with the system, essentially. The problem with this is simply that the module doesn't appear to have been written to troubleshoot system configurations for syslog. It tests for certain conditions and failure scenarios and provides error messages for such (you can see that from the source code.) However this is limited to only the problems it can detect. Issues further upstream in your system configuration aren't even examined by this module.
After having looked at the source code briefly myself, I don't see anything there which is going to help you troubleshoot your system setup.
I feel your pain, and I'm sorry it's not working for you. Unfortunately I must say that I don't think approaching the problem with Sys::Syslog is the right way to troubleshoot the root cause. Best to work it out from the system level.
Check your firewall rules on all machines, and vimdiff the config files. You'll figure it out, but not by using this module.
$ perl -MMIME::Base64 -e 'print decode_base64 "YWNlQHRvbW15YnV0bGVyLm1lCg=="'