POD: a Briton marries into the Hawaiian royal family circa 1885-1890. He's a scion of an important aristocratic family (brother of a Duke?) and a capable veteran soldier. As such, he bitterly resents the couo d'état by American residents that set up the "Republic of Hawaii". In 1898, while the US is distracted with the Spanish-American War, he leads a monarchist counter-revolution. He has just enough political juice with the British government to get Britain to establish a protectorate...
Another stab at this. Pearl Harbour in a British Hawaii would be a minor base. It might not even have operational radar in late 1941. Second-line aircraft, limited coastal guns and anti-aircraft guns. For various reasons, the British don't think a Japanese raid on Pearl Harbor is a serious possibility.
Japan as OTL wants to seize SE Asia. This means defeating Britain, and also deterring or blocking the US from intervening. OTL, the Japanese answer to the latter problem was to strike Pearl Harbor, where half of the USN's battleships were.
ATL, the USN fleet is in California, out of reach.
But... If Japan seizes
Hawaii, the USN is blocked from sailing west. (Or so it would seem.) Japan even threatens the US mainland with long-range bomber attacks (not explicitly, of course). Since Hawaii is not a US possession, the US won't be at war. (Japan also refrains from attacking the Philippines, Guam, and Wake Island; they can be taken if the US declares war. The US could of course make a huge buildup in the Philippines first, but that would take years.)
Invasion and defense of Hawaii would be a strain for Japan, but the defenses would be much weaker. The Japanese might even form a "fifth column" among Japanese-Hawaiians. Or they could, or think they could, recruit Japanese-Hawaiians as manpower for support of the occupation and troops for defense.
In any case, it is highly improbable that Britain would station a large portion of the Royal Navy in Hawaii. Britain was heavily engaged in Europe, and has other, more important places in the Pacific to cover: Singapore and Australia.