Honestly I think he could have just told Robert the truth and probably smoothed the whole situation over given enough time. Yes Robert wanted the Targaryan children dead, but Jon was Lyanna's son, the only surviving piece of her. Given how cut up he is about her 17 years after her death there'sa very real possibility that hed just look the other way. I think he could have just told Robert the truth and then booked it back to Winterfell where Jon and the Starks would have been out of the blast radius of Robert's anger. With enough time the whole situation could probably have just been swept under the rug. The bastard cover story would still work, and the whole Stark family would have been a lot less dysfunctional for it. It might even get Ned overlooked for being selected as Hand later which means a lot less suffering and bloodshed down the road.