On rereads, my take on the whole incident is that it was not supposed to make us think Sansa was disloyal or a bad person. It was a warning about Robert's incompetence as a king.
First of all, if it was intended to cause readers to view Sansa negatively, it would have been told in an Arya POV. She is the member of the Stark that does not get along with Sansa.
Instead, we are shown the "trial" and death of Lady through Ned's eyes. What this incident does is establish a pattern of behavior in Robert. Robert does not like ruling. He has no interest in making difficult decisions or listen to people's problems. He leaves these things to his council and Cersei, at his own peril.
Robert frequently stands by and allows unjust things to happen. He has the power to change them but doesn't want to. Mycah and Lady are not the only victims of his indifference. He also allows Tywin to have Elia, Rhaenys and Aegon to be killed in an extremely brutal fashion. He does not punish or reprimand Tywin or Gregor and Amory Lorch for the horrific way they were murdered. Robert also allows his council to plan the murder of Daenerys. When Ned explains the immorality of this, Robert brushes him off. Robert also allows Littlefinger to bankrupt the crown because "counting coppers" is of no interest to him.
Back to the topic of Lady. Here are the quotes that support my hypothesis.
Clearly Robert is annoyed that he has to take care of this matter. This is befor the proceedings have even begun.
Robert was slumped in Darry's high seat at the far end of the room, his face closed an sullen.
"Why was I not told that my daughter had been found?" Ned demanded, his voice ringing. "Why was she not brought to me at once?"
He spoke to Robert, but it was Cersei Lannister who answered. "How dare you speak to your king in that manner?"
Robert does not take the initiative to make sure that Ned (who must been frantic with worry) that Arya was all right. Then he allows Cersei to answer for him. Again, we can see that he just doesn't care that much.
When his son was done talking, the king rose heavily from his seat, looking like he wanted to be anywhere but here. "What in all the seven hells am I supposed to make of this? He says one thing, she says another."
Once again, Robert is massively irritated at having to do his job (parenting or kinging take your pick).
"We have a wolf," Cersei Lannister said. Her voice was very quiet, but her green eyes shone with triumph.
It took them all a moment to comprehend her words, but when they did, the king shrugged irritably. "As you will. Have Ser Ilyn see to it."
The two bolded words say it all. Robert did not stop to think about how unjust and cruel punishing Lady and Sansa was. He just wanted it over with. He wanted to get Cersei off his back. It is only after the Starks cause an uproar that Robert comes up with the excuse that being a direwolf, Lady would probably become violent anyway.
All Ned could do was take (Sansa) in his arms and hold her while she wept. He looked across the room at Robert. His old friend, closer than any brother. "Please Robert. For the love you bear me. For the love you bore my sister. Please."
Sansa is in anguish and as a result, so is a Ned. Ned even brings up Lyanna, the girl Robert is supposed to be so in love with. Nothing moves him. Robert is supposed to be Ned's closest friend and he still can't be bothered to stand up to Cersei.
"At least have the courage to do it yourself."
Robert looked at Ned with dead, flat eyes and left without a word, his footsteps heavy as lead. Silence filled the hall.
As loud and gregarious as Robert is, silence might be the word to describe Robert's time as king. He has all the power, but he doesn't care to exercise it when he can speak up and stop an injustice.
The debate always rages on about Sansa's behavior and Sansa's character, but I think this chapter is really an indictment of Robert.