I assume since you're a Check Point employee, you're asking this for a customer.
Log Exporter only exports Security Management logs.
It does not export Gaia OS logs, many of which are exported with syslog in Gaia.
audit.log in particular is not included in this by default.
The standard "Linux" ways to do this suggest using a plugin to audispd to send the information to syslog.
This plugin doesn't exist on Gaia and I presume adding it would be an RFE.
You might be able to use the logger utility to pipe this information to syslog, but I haven't tried this.