Hi Zeus, I new to this and screwed things up with my wife but if this was me I would have a sit down with my sister for a heart-to-heart. This is your sister, dude. If she doesn't have your back in this marriage there's a pretty good reason.
I would ask why she thinks your wife and kids are better off without you and what you can do to change this. She obviously has an opinion. Why not get it? The way I see it you sister is giving her opinion to your wife so maybe you should find out what the hell she is saying. You also may want to find out why she is saying it.
Maybe after you talk to her she can give you insight on what is going on. We can't. We don't know your wife.
I would look at it as a job performance review. If my boss secretly told a co-worker why I'm getting fired I would want the information so I change what I'm doing and save my job.
I don't see this as any different.
You know how to talk to your sister. I would go talk to her. I see this as the best opportunity you've had to hear your wives side of the story since this drama began.