The United Kingdom lacks CATOBAR systems for one simple reason: nuclear power. Catapult launches require large amounts of available steam which is only available on a nuclear powered vessel. Adding the necessary boilers to conventional propulsion would leave no space for aircraft. Odds are also good that the stacks would overly complicate the flight deck layout.
Currently the UK gets around the problem by using jump jets on their carriers, which is hardly economical. Jump jets are expensive enough on a per-plane basis to make the costlier nuclear carriers attractive after a relatively small number of ships. In any case, if the people of San Seriffe can't import military technologies then compact reactors or advanced thrust-vectoring jets are showstoppers.
There is a fair amount of development going into electric catapults, which would eliminate the nuclear propulsion requirement. They're difficult from an engineering perspective, but so were nuclear weapons and we all know those haven't cropped up anywhere else after the Manhattan Project. Be careful here though, because if you postulate enough progress in electric launches to make railguns practical the balance of naval power tips back towards surface ships.
The Future Is Yesterday
Nothing prevents you from building a WWII-style carrier if you don't mind being limited to WWII-style aircraft. Specifically you'd need aircraft that are suitable to tail-hook recovery and short takeoffs, which would limit you to turboprops and helicopters. Obviously it would be hopelessly outmatched against a supercarrier and laughable against land-based air forces, but you would have a fair amount utility for roles such as search and rescue, close air support, recon, etc. Given the advances in munitions and engines since the war you'd likely give any neighbors limited to surface vessels a black eye too. Aircraft are even more of a threat to submarines than they were then, and a modern torpedo bomber won't be dropping an unguided fish.
Winning On the Cheap
Depending on your exact goals tankers or submarines are likely a better investment. Aircraft tankers are essentially a flying gas cans and a straightforward conversion from cargo aircraft. They'd be a much more cost-effective strategy to increase the range of a small number of jets.
Diesel-electric submarines are in an odd spot now. Having to spend time near the surface does increase their odds of detection in the long run, but not having to run the machinery associated with the reactor makes them quieter than nukes when submerged. If your submarine doctrine doesn't consist of trailing enemy boomers for months on end they're really not a bad deal.
One Final Problem
As strictly worded your islanders won't be able to manage anything remotely resembling a naval vessel, because 30m yachts aren't getting it done. The Arleigh Burke is 160m long. Your shipbuilding industry is just too small to even convert cargo and cruise ships. Even assuming a suitable port or temporary dry dock there's the question of cranes, materials, skilled labor, etc. All that infrastructure and expertise just sits around? Lots of people contract out ships, but then they're generally built elsewhere.
The only way it really makes sense is if the country is already invested into shipbuilding for historic or political reasons. Cargo ships maybe be a better bet than luxury yachts because they offer a fertile field for various financing arrangements where a large foreign company gets tax write offs while the islanders get immediate income. If building and operating ships is a large part of the economy it's not surprising that the infrastructure and expertise extend beyond boatbuilding to shipbuilding.