I know it's a longshot, but... just going by headshape alone, so think it's possible That Man's third servant is Bardius? Go back and look at the pictures. They both have long head/necks. You're probably being bombarded with Xrd questions so this is probably a welcome break.
This was a working theory at first, but now it can’t possibly be true, for several reasons:
—-“That Man” and Chronus (of the Conclave) are enemies.More specifically the Conclave consider “That Man” as the highest potential threat to them. So it stands to reason that Baldias would be “That Man’s” enemy as well.
—-Sol Badguy confronted Baldias in the events of Guilty Gear VastEdge at Harden Fort. After defeating him, Baldias appears to have disappeared or been killed, the rest of the Conclave members symbolically place an upside-down teacup at his sitting place in the Tea Room to symbolize their loss of a member.
—-The Conclave has effectively been dismantled, so even if Baldias were to survive, only Chronus would be around. Chronus is currently being detained by Dr. Faust for questioning. (It is possible that the Apostle Libraria still exists within the Cradle, but it is unlikely she survived Sol Badguy’s attack).
—-“That Man” appears to be working in his own ‘private space’ within the Backyard, somewhere that no normal human being can contact him. By extension, only I-No, Axl Low, Raven, and his mysterious 3rd Servant seem to be individuals capable of contacting him.
—-Bedman can be ruled out of the above possibility since Bedman IS considered “That Man’s” enemy at this point and appears to have attacked “That Man” during their last encounter.
—-Dr. Paradigm is a theoretical possibility, but he himself did not meet the Gear Maker until after 2186. ”That Man’s” 3rd Servant has been by his side since 2180, during the “Lightning the Argent” events.
So yes, there are no current leads on the identity of “That Man’s” 3rd servant.
Anji MIto may still be keeping an eye on the Gear Maker, however.
Perhaps we’ll find out in the NEXT game?