Have you called them??
I had a similar issue whilst seeking training while on a long term assignment in th US and travelling under a B1 in lieu of a H1B.
The flight school didn't really know, the Alien Flight programme didn't really know and I ended up flying over schools over the US to find out which gave mixed results. I even phoned the visa issuing authority to ask and they didn't know, saying that I could exchange my visa for a student one if I like and have both but I needed to return to the uk for that.
Eventually I contacted the AOPA as a non-member and the guy at the over end of the phone told me that there was a lot of confusion over this one and that there was no hard rule applied. He could not offer specific advice but in an off the record conversation just chatting he did suggest that even the government didn't know and that as per most of my flight school phone arounds I could get training if it was not my main reason for bring there.
So I applied to the AFSP and was approved and now an FAA CPL holder. Now back home I'm doing the opposite and wondering why H-pilots need to answer questions about flying 7000 miles on great circle tracks, naming the point at which you are after 3500 miles. That's simple for me - I was at the fuel pumps before 500 miles!!
So worth trying to phone the authorities just to check you have the rules straight in my opinion. Good luck with it.