Who is saying BC didn't lie? He's a lying liar that lies.
Are all the lies coming from the custody deposition? Anyone that has been through a nasty divorce and custody dispute knows that affidavits are flying in both directions disputing what are presented as truthful facts. Depositions, or Examinations for Discovery, are very invasive questioning periods where completely unrelated questions are asked. It can be a very unpleasant experience, and I doubt anyone decides to provide all the information that is asked. For example, Brad claims that he doesn't know his nephew's name. I don't believe this. I think he is well aware of his nephew's name, but has no intention of dragging a preschooler into his mess.
Another point to consider is that stress interferes with memory. If Brad was under stress, he would have naturally had difficulties keeping everything straight in his head.