Tri Bike or Wheel Upgrade?

Short answer: tri bike.

Long answer:

To answer your question, you have to answer "Can I ride aero on my road bike now for long distances"?

When you get right down to it, the major advantage of the tri bike geometry is to enable the rider to stay in the aero position comfortably for hours.  The instant you sit up on a tri bike, that advantage disappears.  In fact, sitting up on a tri bike may be less comfortable than riding a road bike, so you may actually ride slower over distance. 

The "saving legs for the run" bit is a well dispelled but oft repeated myth, BTW. Let's not go there.

Thus if your answer is "yes, I can ride most of the HIM in aero on my road bike with clip-ons", then you won't get much of an immediate advantage from a tri bike alone.  You would get it from a tri bike with aero wheels, aero helmet, etc - the full package - but not from a tri bike alone.

If the answer is "no, I can't ride aero for most of the HIM", then adding aero wheels won't do much for you.  Your body sitting upright slows you down far more than non-aero wheels, so getting a tri bike so you can go aero would do you the most good.