i would absolutely stay the entire time on BI.
Drive up to Mauna Lani and do some amazing snorkeling at the Makaiwa Bay. Have lunch at Napua right at beachside. I realize you're staying at the FS which is amazing, but their beach is nothing to write home about. Rocky and not what you picture a hawaiian beach as.
Drive down to Kailua one day and see the Pu'uhonua o Honaunau historical park. Keep your car there and walk over to snorkel at Two Step next door. (Can you tell I focus on Snorkeling - it's where the BI really shines).
There is so much to do on the BI and you lose so much in transit days going between islands - i feel staying on BI will allow you a more relaxed pace and a chance to really soak it all in.