I'd say for a short trip, vacation and some hunting, goats would be the best bet. For axis deer or mouflon, I'd suggest staying from Sat-Sat, on Lanai the full 8 days of bowhunting.
There are other options but most would involve hunting on private land.
Let me know when you start to narrow things down. Just remember although Oahu got some nice beaches and hikes, it is really crowded!
Start doing some searches on google...