It depends a lot on how fast you hope to do the ride. Is this a multi-day ride with a sleep stop in the middle, or all in one go? Is there a time limit? If your goal is just to finish, then double century training plan is probably fine as far as fitness goes.
The big differences between a double and a triple are not really fitness issues. One difference is that while a lot of people can finish a double within the hours of daylight, a triple will require some night riding (assuming you're doing this all in one go). Also, most people can finish a double without sleep deprivation becoming an issue, whereas it may become an issue in a triple. So it's good to have some experience with night riding in a sleep deprived state. One way to train for that is to do a century ride starting at 9pm. It's a good way to make sure your lights work, you can navigate in the dark, and you can deal with being awake on a bike all night.