TimT
I made the same trip last May, stayed at the same hotel on a business junket as well.
The Francic Lil Brown courses on the premises are both very nice resort courses with a couple of great holes on the water, not too tough no matter how far you go back. When I was there the North course was undergoing renovation with holes out of play, so make sure you check the status before booking your times.
I was able to play Hulailia and enjoyed it, I am a Nicklaus fan so take the recommendation with a grain of salt.
Hapuna was excellent as well, of the courses I played on the Big Island it was probably the best test all around and in the best condition.
If you can figure out a way on, Kukio, the private course on the other side of the road overlooking Hulailia is supposed to be the best course on the island, although I did not play it, I did play Hulailia with a couple of seasonal residents who raved about the course, the conditioning and the ambience.
I did not play Mauna Kea, at the time it was supposed to be in very poor condition.
As a side note, the Fairmont is not bad, but definitely eat dinner at the Four Seasons on the beach, great food and a great setting. Enjoy your trip.
Jeff