bookworm
Certainly one gets that credit for one's IMCR training, that's not a problem.
I had a conversation recently with someone who could not quote chapter and verse but there appeared to be a legal reason why any FAA license/rating training required an N-reg aircraft.
However this could just be one of the countless OWTs going around GA. I suspect this is the case because 2D wrote recently that any IR instructor (instructor rated to teach an ICAO IR) will do for the FAA IR, all of it except the checkride.