Royal Hawaiian Army --- Uniform and Equipment: 1941-1943
https://turtledove.fandom.com/wiki/Royal_Hawaiian_Army <--- Wiki article
King Stanley Owana Laanui was able to raise a small army with he was in power. Although ceremonial at first, mostly used to garrison Iolani Palace, during the US Invasion of Hawaii in 1943 the army saw action against US soldiers while fighting along side Japanese troops. Their performance was mixed, with units either surrendering without a fight or battling US soldiers in tenacious combat to the death.
At roughly battalion strength, the Royal Hawaiian Army may have been somewhere between 300+ to 800+ men, divided into multiple companies. They were said to be armed with captured Springfield 1903 rifles, but were not issued ammunition since they were, for the most part, used for ceremonial duties and due to Japanese mistrust. It was not until the US invasion that Royal Hawaiian Army was issued ammo in the coming fight.
I haven't read that far yet, but for now I'd be willing to take a guess at the equipment and uniform of the Royal Hawaiian Army. This will exclude a possible ceremonial uniform and focus more on a practical combat uniform.
^^^ --- I think the uniform might be the old summer service uniform used before the occupation. Due to the logistical restraints on the Japanese, I imagine the Royal Hawaiian Army would be issued with surplus/captured US equipment and uniforms. This would include using the Brodie helmets as well. All US markings would have to be removed and perhaps simple insignia or armbands might be used to identify Royal Hawaiian units.
^^^ --- As such they would most likely be kitted out with the M1903 Springfield and ammo belt, removing any US markings.
^^^ --- I do think, however, that the Japanese Army might issue the Royal Hawaiian Army with Japanese equipment as well. This would help if only to identify the Hawaiians as being on their side and not mistaking them for US soldiers. They could probably issue them with Japanese caps, helmets, and ammo belts.
^^^ --- Also likely is the Japanese Type 99 Arisaka being issued to the Royal Hawaiian Army, if in sufficient quantities.
As such, I think we would be looking at a Hawaiian Army that is kitted out in pre-invasion fatigues with a mix of Japanese and US weapons. I believe that with the logistical constraints on the islands by 1943 that the Hawaiian Army would be kitted out in a very simple manner, with identifying armbands used to denote them, if at all.