Most of the aircraft carriers during WW2 where small ones often as small as 8000-12000 tons.
These were low cost aircraft carriers, defensive carriers, not offensive ones, used often for escort.
The Soviets also imagined a defensive small carrier - to give their subs enough time and distraction to get them into the deep.
Since such WW2 aircraft carriers only used prop aircraft, relatively light and small - the deck and general structure did not need to be very strong. They were simple ships relatively, some converted in fact from merchant ships!
If PN built one and had Super Tucano type prop aircraft - one could easily look at a 200 million USD "aircraft carrier" at about 8000 tons and 11000 fully loaded.
If you built a Super Tucano type aircraft with rotary engines or piston engines, you could get aircraft on board at abou 4 to 5 million USD per aircraft.
Incidentally, the main advantage of turboprops over piston / rotary is that turboprops are more efficient at over 5000 feet. But since in maritime combat, such aircraft would be flying relatively low anyways, this is actually a cheap and effective solution. Except the more frequent need for overhaul, which isn't that bad if you look at the numbers and the cost.
Now, if you had this "pocket" aircraft carrier, you could project power - have AShM and AAM armed Super Tucano types that could play a meaningful role in fleet air defence, and anti surface, as well as for gunboat diplomacy in the Middle East and Africa.
For instance, in a Yemen - like situation, you could bargain with some oil sheikhs for very favorable goodies / influence in return for bombing some cave men. These aircraft would be perfectly suited for such a task.
Imagine 12 fighters (admittedly prop fighters)
4 helicopters
12 UAVs.
Ability to deploy 100 - 300 troops.
At the cost of below $300 - $400 million.
Remember, such a ship would not need any of the advanced armaments that make a frigate or a destroyer expensive - no OTH radars, no AShM missiles, no main gun, no .... etc. Jus a basic radar, a flight deck and space. More or less.
