Unless I’ve missed something. He doesn’t claim that he had to go to training before his father’s funeral. He claims that the possibility of him doing so came up in conversation.
Without a recording of that dialogue, we can’t determine if the claim is true. He is publicising his book though, so has to throw shiz at somebody to create an interest.
If the episode is reported in the same context in the book, then I imagine that Moyes will inevitably, either be asked for a response, or will challenge the facts directly, if he has a different recollection.