I just can't see Wilson breaking Washington precedent without hellish good reason. And even if he decide run third term, it would be pretty much of uphill fighting. Firstly him should assure other Democrats why him should run third term instead someone else Democrat. And even if the party agrees, he would face much accusation and tainting campaign being disrespectful towards Washington and they might even blame him being powerhungry despot, perhaps comparing him to Julius Caesar.
Wilson had the ego, the hubris and the blindness of conviction to attempt it. But I agree, he still needs a push, the tradition was a strong barrier, but in his heart Wilson would not simply let a tradition stop him if he found a reason that moved him.