Aloha,
Ruh-roh........through no fault of yours, Rebecca, you've just pressed a nuclear warhead button.
All the major car rental agencies prohibit driving the "back road". See here for the map:
http://www.honolulupublishing.com/MauiMaps/maui_map_island.html
This topic comes up often and nearly always results in a war between those who say don't and those who say do. You might want to do a search for some past threads and you will see what I mean.
Also, your friend is actually not helping your vertigo. If you did the back road, you would be on the outside, with no guardrails for nearly the entire road until you reached just outside Hana.
From Hana back to Kahului, you would be on the outside too (as you will be if you just drive the permitted route up and back).
Going the 'legal' route only puts you on the outside once. You will be on the inside on the way to Hana.
I think you should watch some videos of the road. Many of the one lane bridges are suspended over gulches, and there are 54 of them. There are 617-649 hairpin curves on the road (depending on who counted).
Your vertigo may make the whole idea moot, not just staying in Hana but even driving all the way there.