Disagree. This is like saying that you have to “drive to each part of a state” to say that you drove in that state. (Case in point: do you require driving on every island found in a state?)
I thought there were ferries that could take you from one island to another w/o the need to “ship” the vehicle.
Or wait until you were thinking of selling it anyway and then just sell it in Hawaii and buy the new(er) car on the mainland so you only have to ship it one way.
Alternative: join a branch of the military and get yourself stationed in Hawaii (not trivial - my brother had to volunteer for Korea first in order to line up a Hawaii PCS) and the military will ship your car for free! on the taxpayers’ dime.
Is the Alaska trip going to be like the remainder of your road trip: spend 4-5 days driving to the border, and turn around after you cross the state line?
I saw this elsewhere and it reminded me of the CS road trips: here is a route map for every city in the US named Cleveland. Could be the next CS road trip.