We lived on the Big Island 20 years ago, and NO, even local produce was not less expensive.. Local eggs cost more than those shipped from the mainland--I never understood that! Some local produce was shipped to the mainland (like pineapples) and then shipped BACK to the local Safeway where it was overripe and very expensive!
Farmers markets were just getting started back then, but I hear they are a good affordable source of local produce. And get to know your neighbors! People who have an abundaance of papaya, guava, breadfruit, etc. are eager to share. Gardens grow fast. We saved a bunch one summer when birds "planted" a cherry tomato plant in our yard. Our neighbor had 40 fighting cocks and a few hens--eggs came from there. Meat will be costly, even if there are beef cattle on the Big Island. You can shop sales for fish at local markets--be adventurous and eat what's inexpensive--ask how to prepare things and don't fear raw fish. Neighbors also fished and would sometimes share the bounty.
I did home health there, and once was scared out of my wits by a tied up wild boar my patient's family caught--it was going to be dinner that night. I'm not suggesting hunting, but that's how some resourceful people subsist there! And they forage for things like fiddlehead ferns, tiny shellfish (opihi) and sea vegetables, too.
We had a tight budget. We made what we could from scratch, and mainland family sent coupons which didn't get published in the local paper. In those days we ate some packaged foods and it helped with paper goods and cleaning supplies, too.
The cost of living in paradise is steep!