I did 4 days on the BI with the little (no relations) a few weeks ago.

We flew into KOA, rented a car, spent a day & a night in the area.
We kayaked out to Captain Cook Monument, snorkelled with the honu at Kahaluu beach, and went down to the Heiau. (that was awesome; we arrived ~4:30 and had the darn place all to ourselves. Walked way out on the lava, saw some old fisherman's memorial and really enjoyed every minute of it. ^)

Then we drove around to VNP (right thru TS/TD Cosme) and spent 2 days & a night there.
Then went into ITO, overnighted there, turned in the car and flew back to HNL.
(note: don't plan on doing much in ITO on a Sunday; many places, including museums, are closed Sunday)(and it will rain )

Find out if you can fly to one side, rent your car, return it on the other side & fly out from there.
Good luck!