I was in a similar situation at one time, on a J1N exchange training visa. The immigration attorney advised me that it wasn't acceptable for me to do IR training as it wasn't defined in my J1 programme, and as you could only hold one visa, there was no work-around.
I subsequently held an H1B which doesn't have that restriction although you still need the TSA approval.
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.