I had a similar experience last November, de-boarding and clearing customs in under an hour.
The Otter (the prop plane island jumper) makes several back and forth trips through the three islands per day, so I wouldn't worry if you miss your scheduled flight. They will just have you wait for the next arrival. In fact I think my Otter ticket didn't even have a scheduled flight time. It was more like they just put me on the next plane after my arrival.
They do have to shut down operations to Little Cayman when it gets dark because the runway has no lights. But if you get in around noon or 1, there will still be several shuttle flights after that.
Also, just a heads up for you, you can only carry on items on the Otter that fit under your seat. They are bench seats, much like those in a school bus and there's not a lot of room. The rest of the luggage goes in forward and aft compartments on the plane. It's not a big deal as long as there are not a lot of people on your flight. If the plane is full (like 15 passengers) your luggage may have to wait for the next flight, which could mean an hour or more sitting on the tarmac. Just keep that in mind. I would consider putting locks on my carry ons and keeping anything that is temperature sensitive (like computers, camera gear) in small bags with me just in case it doesn't make it on your flight.
When you get to LCBR, try to get on the boat with DMs Neil and Sharron. We were on two different boats last year with different DMs and Neil and Sharron were by far the best.
Let us know how the resort is looking. We were evacuated from there 2 days before the end of our trip due to Hurricane Paloma. I know the Brac got hit really bad and LC took a beating too. We also had an issue with our luggage not making it on the Otter on the way out during the chaos of the evacuation and were without all our dive gear and clothes for 2 weeks. We got everything back in the end though.
Have fun and feel free to PM me or ask here with any questions.
Oh, and here's a shot from inside the Otter to give you an idea of what to expect.