I just had a chat with a trusted friend who advised me to go ahead and pursue the training with the FIL. I'm still his son-in-law, right? It's my GAL, darn it. So he agreed that I should go.
Also, he suggested that while visiting, I broach the subject of my wife's relationship with her boyfriend. Specifically, for the purposes of finding out their level of support of a woman who refuses to terminate an adulterous relationship, pursue a divorce, and become wife #4 for this guy.
Several months ago, I told the in-laws that I will never ask them to do anything or not do any thing, and that will not change. However, as things stand now, I believe that I have very little to lose in opening up this discussion with them. They are getting only half of the story (and I suspect a lot of lies: I am a bad guy, she has broken off relationship, there is no hope for us, etc.).