It doesn't seem entirely clear what additional activities you can do on a J visa, so it may be worth getting an updated view on the matter.
Who can answer definitely answer this - should I reach out to an immigration attorney?
Sorry, J1 qualifies as Visiting Scholar for the US and your purpose is to teach the US guys whatever you know. It does not qualify you to undergo training. I ran through this a couple of years ago and we found definitely no way to do that legally.
That's just sad - I'll just have to accept it if there's no other way.