After reviewing dozens of places on the Big Island and reading reviews with most mentioning cockroaches, we settled on Paul's place which had 221 reviews at the time but none mentioning any bugs. After staying at Paul's place I came to the conclusion: there is nowhere on the Big Island that you can be assured to not see cockroaches. We saw a large one on day one and a few small ones throughout.
Although perhaps unavoidable, seeing cockroaches leads to an unusually thorough examination of the property's cleanliness, which didn't bring much further comfort.
Paul did offer a full refund, but we chose to stay after being uncomfortable with the idea of going somewhere else and still seeing cockroaches. Other than that the property had a good location with comfortable parking, a communicative host, good guest information and was large and spacious with double sinks.