Once you get to the club, Only-He-Stands-There suggests a deal. If you return to Ranis Athrys, and tell her that he's stopped offering training, he'll train you (prior to this quest, he refuses all services to the player). You can either accept this, kill the Argonian, or use bribes and Speechcraft to get his Disposition high enough so that he'll just stop offering training. If you kill him, you are rewarded with four Scrolls of Elemental Burst:Fire and receive an extra five points of disposition with Ranis.
Choose whatever method seems to work for you, and then return to the Mages Guild to speak to Ranis Athrys.
Only-He-Stands-There may not give options after explaining why he is offering the training, therefore the quest seemingly can only be advanced if you kill him.
The quest can be advanced by going back to Ranis at a later time and asking her about duties again. She will offer the same quest again and your journal will update.
The following Journal ID and Index codes can be used with the Journalconsole command to manually update the quest to a certain point.
Unsanctioned Training (MG_StopCompetition)
Index
Finishes Quest
Journal Entry
10
Ranis wants me to convince an Argonian at the South Wall Cornerclub to stop offering unsanctioned training in Restoration.
50
I convinced Only-He-Stands-There to stop offering unsanctioned training.
70
Only-He-Stands-There agreed to train me in Restoration. In exchange, I am to tell Ranis Athrys that he is no longer offering training without Guild permission.
100
Finishes quest
I told Ranis Athrys that Only-He-Stands-There wouldn't be offering training anymore.
110
Finishes quest
I told Ranis Athrys that Only-He-Stands-There wouldn't be offering training anymore. In exchange, he will train me in Restoration.