How to forward specific log file outside of /var/log with rsyslog to remote server?

Just setup an imfile rule in your /etc/rsyslog.conf

$ModLoad imfile
$InputFileName /data/mysql/error.log
$InputFileTag mysql-error
$InputFileStateFile stat-mysql-error
$InputFileSeverity error
$InputFileFacility local3
local3.* @@hostname:<portnumber>

This watches a file and saves to the local3 facility in syslog. Then you can send all data from the local3 facility to your remote server. You may also want to add the following to your rsyslog conf (usually /etc/rsyslog.d/50-default.conf on Ubuntu) to not save the local3 facility to /var/log/syslog:

*.*;auth,authpriv.none,local1.none,local2.none,local3.none,local4.none,local5.none,local6.none          -/var/log/syslog

Additionally, I would encourage some reading from the following rsyslog docs for more advanced filtering:

  1. The Property Replacer
  2. Filter Conditions