The unfortunate answer to your question is "very unlikely."
During the time of Cook's voyages (1768-1779) every naval power on earth (the Dutch, the Portuguese, Spain, and of course, England) was trying to map the world. Not just for the purpose of business, but for military and political purposes as well. Naval technology, optical technology, navigational technology, and cartographic technology had all come to a point where the world took the next big step forward.
Yes, Hawaii is small, and it would seem like it was difficult to find. However, Cook was specifically a skilled navigator and cartographer. I can't find how tall the HMS Resolution's mast was, but it was not uncommon at the time for mast heights to reach 300 feet. That would allow them to scan a radius of over 20 miles at any time (depending on weather, or course). With an average speed of 5 or so knots (about 5.5 miles per hour) and, say, a 10 hour day (55 miles) that means you could view about 2,000 square miles a day.
But Cook wasn't depending on eyesight alone. Wherever he stopped he would talk to natives about their neighbors, both myth and reality. He wasn't just doing this for alturistic purposes, every inhabited island represented resources and supplies a ship at sea could use to stay at sea — especially during war. So he hopped from island to island, slowly charting everything he found.
Until he found Hawaii.
Considering how much effort was going on at the time, the odds of not finding Hawaii within the period of 1750-1850 are, frankly, zero.
Now, having said that, I wouldn't just throw away your ideas. Cook was met with enough equanimity, curiosity, and hospitality to allow for Hawaii's then dependent development. However, the Sentinelese are a stone-age people still living today. Despite the island being very, very close to the coasts of Burma, etc., the people survived cultural contamination — basically by being the meanest mother hubbards on the planet.
Regrettably, this would mean changing the basic nature of the Hawaiian people such that they beat off Cook and everybody following such that people stopped wanting to land on Hawaii. The island would still be discovered (couldn't be avoided), but the people would have developed more-or-less independently.