A cab is going to be the easiest way to get there but not the cheapest. The problem with the off site hotel shuttles is their frequency and their pickup / drop off times.
If you can get to the TTC using the shuttle, you still have to get to the Magic Kingdom to transfer to a Disney bus to get to the Lodge. You are looking at about an hour each way (it is less time if you don't have to wait for a bus) and then have to make sure that you can catch a shuttle back to your hotel.
There is no monorail service to AKL.