Does the Bavarian Illuminati make a privileged attack in addition to their two actions?
No, the Bavarian Illuminati's special attack is not a free action.

It is not listed as an example free action in the list on page 7 of the rules.  It's function is built into the regular attacks used.  Note that the only mention of the special power in the rules section is inside of the Interference section on page 7.

The Bavarian's privileged attack is simply a way for them to avoid paying the usual cost to make an attack privileged. Instead of Discarding a special card, they can spend $5 once/turn to make an attack privileged.

This does not mean that the groups participating in the attack are exempt from the general rule that each group may only participate in one attack per turn.