Justin was an ex Warden and knew Bob was on the most dangerous artefacts list (below the Eye of Balor, The Spear, The Shroud, The Crown, The Placard and Excalibur) which in no way should be in the possession of an individual wizard. Ever. So he didn’t tell anyone in the White Council.
He may have told whoever he was working for but the people he was working for weren’t very nice, abducting children, so I doubt he told them about his Ace in the Hole
That does leave a bit of a plot hole, because supposedly Justin told no one he had Bob because as Luccio told Harry in Small Favor I believe, it was illegal to possess Bob and that he should be destroyed because if I remember correctly no one outside of the Host of the Archive should have access to that much information. Harry knew about Bob and perhaps Elaine might have as well because he eavesdropped when Justin was getting information from Bob.
Elaine could be suspect because she might be infected with Nemesis, the general theory is she carried the infection to Aurora when she fled after Justin died. We've seen how hosts infected by Nemesis act, assuming that Cowl is playing for the same team as Nemesis, it could very well have been her that told Cowl. Another simpler explanation is Cowl followed Harry or had Kumori follow Harry after their encounter in Dead Beat because he thought Harry had or knew the where a bouts of Kemmler's book, which he also wanted so he could do Darkhallow. Harry took Bob with him to Murphy's house, Cowl followed, saw the skull and immediately knew what it was, and took Bob.