I apologize that I don't completely understand your question, but I think you're asking about whether the moment of inertia tensor would change as seen in a non-rotating frame. Yes, you are correct that the tensor would evolve with time in a frame that was not rotating with the body. That's the advantage of using the body frame (i.e., the frame that rotates with the body) -- in that frame, the tensor does NOT change with time. But that frame is non-inertial, so you have to include the non-inertial effects when you use that frame.