I’m not expert on this issue but it’s my understanding that the reason why U.S. spec. 370s don’t have a functional real fog light is because the fog light provided for Euro and Can spec vehicles don’t confirm to U.S. regulations.
Now, as a practical matter, I doubt anyone would get pulled over by a LEO if they are using a rear fog light as a true rear fog light (that is, turned on in foggy/low visibility conditions, etc.). However, I would be cautious about using the rear fog light as an additional brake light or other use that may attract attention you don’t want.
I've got the J-spec light and am waiting on a switch from Japan (to mount in the center counsel in the empty space between the two heated seat switches) so that I can turn the light on/off manually as desired...hopefully that will work out as intended.