Casio says no. The real question is when was the first NCC constitution class commissioned? In the mirror universe Enterprise episode the NX Defiant a constitution class prototype was pulled into the mirror universe from the prime time line. Makes me wonder.
Edit: Ooops bad memory. It was the NCC 1764 pulled from stardate 2269, 114 years in the future.
Could making the NX refit into a "skin" for the current NX be a good compromise?
Errr no.. supposedly the Refit was more powerfull than the NX class but theres not mmuch to go on since it was only seen as a picture in "Ships of the line". i would look at it and go EH and stick to my NX class.