An island of that size will have been widely known of for the last hundred years or more. Which means that it will be documented in hardcopy atlas, shipping maps, aircraft maps etc extensively. Ships and aircraft travelling to the island know what coordinates they need to travel to and how long it will take from any given port. Ships also need to know what shipping lanes are safe - they are going to know that one big island has vanished and a new one has appeared. So the rest of the world will know almost immediately, as will any citizens of the island travelling to anywhere else. ("Hey Marge, how come it takes 90 min to fly to in a Cessna these days and 30 min to fly to when it used to be one hour to either of them?")
There will also be issues for fishermen (professional or amateur), divers, tour boat operators and others who know what the reefs and sea floor around the island are like. This expands the scope of the problem, as even if it is possible to move the land surface of the island there is also a requirement to move the entire seabed etc around it for probably several kilometres radius.
Finally, most modern mobile phones, smart watches, small boat navigation systems etc have GPS built in - suddenly all of the mapped coordinates will be way off. Unless the population is composed entirely of subsistence farmers who cannot afford any of these devices, the general populace will know they have moved.
There has never been any attempt to move an entire island because it is well beyond current or even near-future engineering capabilities. The best that the organisation could hope to do would be to reproduce the original island at a new location. This will truly take the unlimited budget advertised - it will take trillions of dollars and decades of time just to reclaim the necessary amount of land based on experience in other reclamation projects like this one.
Best approximation possible
The nearest it would be possible to get to the desired outcome would be:
- Spend the unlimited budget reproducing the island at a new location 100 km north. This cannot be hidden from the world, there will need to be a cover story stating that it is for an isolationist cult. (Good luck dealing with the environmentalists who will go berserk at someone trying a massive reclamation project.) Remember that you need to sculpt not only the island, but also the surrounding reefs, seabed etc - good luck with that. Reproduce all structures, then devastate them in a way that will simulate a massive storm (see below).
- Wait for a big hurricane / cyclone (depending on which side of the equator the island is on). The media is in your pocket so you can beat up the likely local effects and the international webfeeds are less likely to be concerned with your little island. Tell everyone that the entire island is going to be flooded/devastated.
- Evacuate the entire population of the island onto ships (hire some cruise ships?) that will be sailing to an area of the Pacific outside of the storm's path. Jam all GPS signals as soon as everyone is aboard.
- Get the greatest removalist team of all time to grab as much stuff as possible (that you don't want to have been destroyed by the storm) and move it to the replica to the north.
- Repatriate the entire population back onto the Mark II island 100 km to the north, blaming all differences between what was there and what is now there on the storm damage. Continue jamming all GPS signals and practice intensive censorship of all communication with the outside world (earmark lots of the budget for this) - blame this on Google or pick another target to demonise. Ban all overseas travel based on a fictitious quarantine threat from the rest of the world. The island must continue international trade (too small to be self-sufficient in a technological society) but remove all personal contact between overseas shipping crews and local populace.
- Make up a cover story for the rest of the world regarding why the populace are becoming so isolationist post-storm. Do not make up a quarantine threat because this will attract lots of attention from the CDC etc.
It will not work - someone will make contact with the outside world or make star observations or something - but this is the closest possible on Earth today.