I always suspected that Ishy had ordered a hit on Bart because he helped out Fain, but when I think about it the Lanfear angle actually makes sense.

Bart is killed pretty quickly in the book. Ishy could have struck that quickly, though it would mean he learned about Bart's actions pretty quickly. It's possible Ishy wouldn't have killed Bart if he had learned that Fain had given him fraudulent orders, since Fain does rank rather high in the DF hierarchy. Also, chessmaster that he is, Ishy might have been able to just adjust his plans to what Fain was doing, and Bart might have been important to whatever plans he had for Cairhien.

Lanfear OTOH, was closely involved with the events surrounding the Horn, and she was trying to use it to tempt Rand. I don't think she cares so much about Fain going rogue, but rather she flew into a rage when she found out he had taken the Horn through the Ways, disrupting her plans. And Lanfear has always taken bad news from DFs...badly. I think it makes sense that she went to question Bart, found out what happened and literally ripped his head off.