Boy, that is a tough one. I prefer the
Spirit of Aloha Show, my wife prefers the
Hoop Dee Doo, and plenty of people prefer
Mickey's Backyard BBQ, largely because it has entertainment and character experiences.
But here is why I recommend the Aloha Show above the others.
1. The BBQ is seasonal and outdoors. If you have only one experience, better not to choose one that might get canceled by the weather.
2. While the entertainment at both Aloha and Hoop Dee Doo are fun, and the food at Hoop Dee Doo might be a bit more kid-friendly, the Aloha offers an exotic experience that you may never get in your life, unless trips to Hawaii are part of your likely travel plans in the future.
3. It is easier to get to the Aloha show than Hoop. You take transportation to the Magic Kingdom, and then monorail to the Polynesian. This could be more important than you might imagine, depending on where you are staying.
But I recommend you view their pages (linked above) and consider them as all great options, and only use my comments as a possible tie-breaker.
Have a delicious vacation!