I can't speak for Einstein or Hawking, but the MATH doesn't lie, and the underlying logic is pretty clear on this matter. If we're flying towards each other, we will both look at each other and we will be able to say the same three things: "You're moving towards me, you have more energy than me, and time is moving slower for you."
I think that's backwards. If we're moving toward each other we're both blue
shifted, not red shifted.
Doppler shift is immaterial. The speed of light is still constant in both reference frames, and time dilation is a consequence of this.
Although, I admit it's been so long since I actually did the equations that I can't remember anymore if a positive or negative integer for velocity would actually make a difference. I remember thinking that it should, and being told by someone smarter than me that it wouldn't, but that same person suddenly went on to explain how the GPS system is more proof of special relativity.
I'll probably sit down and play with the numbers later tonight, just to be sure.