Aha !
The FAA told me that training is permissable providing you have a full FAA licence & that the a/c MTOW is 12,499Lbs or less . My situation is different in the sense that I hold a visa , although it is nearly 3 years old & I've had no additional background checks since issue - seems like a case of closing the stable door ....
The discussion with the FAA was as recently as last Friday .