You cannot be "done" for driving without insurance so long as you have a policy, even if you are not complying with the conditions of that policy. There is a clause somewhere in the insurance legislation which says that an insurer may not refuse to pay a third-party claim even if you were doing something illegal at the time. It's quite logical in a way, since of course many claims result from some transgression of the law (speeding, driving too close, running a red light etc.).
Strictly speaking though, after the 1 year allowance is up your U.S. license is no longer legally valid, so technically you could be charged with driving while unlicensed.